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Review of Functions
Mathematics 53
Institute of Mathematics - UP Diliman
8 November 2012
Math 53 (Part 1) Review of Functions 8 November 2012 1 / 69
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Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
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Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
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Functions
DefinitionLet X and Y be nonempty sets. A function f from X to Y, denoted f : X → Y, isa rule that assigns to each element of X a unique element of Y.
X: domain of f , denoted dom fY: codomain of fThe set of all elements of Y that are assigned to some element of X is therange of f , denoted ran f
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Functions
DefinitionLet X and Y be nonempty sets. A function f from X to Y, denoted f : X → Y, isa rule that assigns to each element of X a unique element of Y.
X: domain of f , denoted dom f
Y: codomain of fThe set of all elements of Y that are assigned to some element of X is therange of f , denoted ran f
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Functions
DefinitionLet X and Y be nonempty sets. A function f from X to Y, denoted f : X → Y, isa rule that assigns to each element of X a unique element of Y.
X: domain of f , denoted dom fY: codomain of f
The set of all elements of Y that are assigned to some element of X is therange of f , denoted ran f
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Functions
DefinitionLet X and Y be nonempty sets. A function f from X to Y, denoted f : X → Y, isa rule that assigns to each element of X a unique element of Y.
X: domain of f , denoted dom fY: codomain of fThe set of all elements of Y that are assigned to some element of X is therange of f , denoted ran f
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Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}codomain: Y = {0, 1, 2, 3, 4}range: ran f = {0, 1, 4}
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Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}codomain: Y = {0, 1, 2, 3, 4}range: ran f = {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
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Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f
= {−2,−1, 0, 1, 2}codomain: Y = {0, 1, 2, 3, 4}range: ran f = {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
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Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}
codomain: Y = {0, 1, 2, 3, 4}range: ran f = {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
![Page 12: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/12.jpg)
Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}codomain: Y
= {0, 1, 2, 3, 4}range: ran f = {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
![Page 13: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/13.jpg)
Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}codomain: Y = {0, 1, 2, 3, 4}
range: ran f = {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
![Page 14: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/14.jpg)
Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}codomain: Y = {0, 1, 2, 3, 4}range: ran f
= {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
![Page 15: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/15.jpg)
Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
−2 7−→ 4−1 7−→ 10 7−→ 01 7−→ 12 7−→ 4
domain: dom f = {−2,−1, 0, 1, 2}codomain: Y = {0, 1, 2, 3, 4}range: ran f = {0, 1, 4}
Math 53 (Part 1) Review of Functions 8 November 2012 5 / 69
![Page 16: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/16.jpg)
Functions
If x ∈ X, the symbol f (x) denotes the element y ∈ Y that is assigned to x.
A function may be written as y = f (x)x: independent variable
y: dependent variable
Alternatively, a function f is a set of ordered pairs (x, y), where
(x, y) ∈ f if and only if y = f (x)
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![Page 17: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/17.jpg)
Functions
If x ∈ X, the symbol f (x) denotes the element y ∈ Y that is assigned to x.
A function may be written as y = f (x)x: independent variable
y: dependent variable
Alternatively, a function f is a set of ordered pairs (x, y), where
(x, y) ∈ f if and only if y = f (x)
Math 53 (Part 1) Review of Functions 8 November 2012 6 / 69
![Page 18: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/18.jpg)
Functions
If x ∈ X, the symbol f (x) denotes the element y ∈ Y that is assigned to x.
A function may be written as y = f (x)x: independent variable
y: dependent variable
Alternatively, a function f is a set of ordered pairs (x, y), where
(x, y) ∈ f if and only if y = f (x)
Math 53 (Part 1) Review of Functions 8 November 2012 6 / 69
![Page 19: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/19.jpg)
Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
The function f may be written as:
f (x) = x2 or y = x2
f = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}
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Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
The function f may be written as:
f (x) = x2 or y = x2
f = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}
Math 53 (Part 1) Review of Functions 8 November 2012 7 / 69
![Page 21: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/21.jpg)
Functions
ExampleConsider f : X → Y defined by the rule
x 7−→ x2
where X = {−2,−1, 0, 1, 2} and Y = {0, 1, 2, 3, 4}.
The function f may be written as:
f (x) = x2 or y = x2
f = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}
Math 53 (Part 1) Review of Functions 8 November 2012 7 / 69
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Real-valued functions of a single variable
Real-valued functions of a single variable:
Codomain: R
Math 53 deals with functions whose domain and range are subsets of R.
If the domain is not explicitly specified:
Domain: dom f = {x ∈ R | f (x) is a real number}
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Real-valued functions of a single variable
Real-valued functions of a single variable:
Codomain: R
Math 53 deals with functions whose domain and range are subsets of R.
If the domain is not explicitly specified:
Domain: dom f = {x ∈ R | f (x) is a real number}
Math 53 (Part 1) Review of Functions 8 November 2012 8 / 69
![Page 24: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/24.jpg)
Real-valued functions of a single variable
Example
1 f (x) = x2
dom f = R
2 f (x) =x2 − 2x− 3
x + 1
dom f = R \ {−1}
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Real-valued functions of a single variable
Example
1 f (x) = x2
dom f = R
2 f (x) =x2 − 2x− 3
x + 1
dom f = R \ {−1}
Math 53 (Part 1) Review of Functions 8 November 2012 9 / 69
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Real-valued functions of a single variable
Example
1 f (x) = x2
dom f = R
2 f (x) =x2 − 2x− 3
x + 1dom f = R \ {−1}
Math 53 (Part 1) Review of Functions 8 November 2012 9 / 69
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Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0(x− 3)(x + 1) = 0
x = 3 or x = −1
The only zero of f is x = 3.
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Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0(x− 3)(x + 1) = 0
x = 3 or x = −1
The only zero of f is x = 3.
Math 53 (Part 1) Review of Functions 8 November 2012 10 / 69
![Page 29: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/29.jpg)
Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0(x− 3)(x + 1) = 0
x = 3 or x = −1
The only zero of f is x = 3.
Math 53 (Part 1) Review of Functions 8 November 2012 10 / 69
![Page 30: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/30.jpg)
Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0
(x− 3)(x + 1) = 0x = 3 or x = −1
The only zero of f is x = 3.
Math 53 (Part 1) Review of Functions 8 November 2012 10 / 69
![Page 31: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/31.jpg)
Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0(x− 3)(x + 1) = 0
x = 3 or x = −1
The only zero of f is x = 3.
Math 53 (Part 1) Review of Functions 8 November 2012 10 / 69
![Page 32: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/32.jpg)
Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0(x− 3)(x + 1) = 0
x = 3 or x = −1
The only zero of f is x = 3.
Math 53 (Part 1) Review of Functions 8 November 2012 10 / 69
![Page 33: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/33.jpg)
Zeroes of a function
DefinitionA zero of a function f is a value of x for which f (x) = 0.
Example
Find the zero(es) of f (x) =x2 − 2x− 3
x + 1.
x2 − 2x− 3x + 1
= 0
x2 − 2x− 3 = 0(x− 3)(x + 1) = 0
x = 3 or x = −1
The only zero of f is x = 3.
Math 53 (Part 1) Review of Functions 8 November 2012 10 / 69
![Page 34: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/34.jpg)
Graphs of Functions
DefinitionThe graph of a function f is the set of all points (x, y) in the plane R2 for which(x, y) ∈ f .
The graph of a function is the geometric representation on the Cartesian plane ofall points (x, y) that satisfy y = f (x).
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Graphs of Functions
DefinitionThe graph of a function f is the set of all points (x, y) in the plane R2 for which(x, y) ∈ f .
The graph of a function is the geometric representation on the Cartesian plane ofall points (x, y) that satisfy y = f (x).
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![Page 36: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/36.jpg)
Graphs of Functions
Example
The graph of f (x) = x2:
−2 −1 1 2
1
2
3
4
0
The points on the graph of f are the points (x, y) that satisfy the equation y = x2.
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Graphs of Functions
Example
The graph of f (x) =x2 − 2x− 3
x + 1:
f (x) =x2 − 2x− 3
x + 1=
(x− 3)(x + 1)(x + 1)
= x− 3 if x 6= 1
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
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Graphs of Functions
Example
The graph of f (x) =x2 − 2x− 3
x + 1:
f (x) =x2 − 2x− 3
x + 1
=(x− 3)(x + 1)
(x + 1)= x− 3 if x 6= 1
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
Math 53 (Part 1) Review of Functions 8 November 2012 13 / 69
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Graphs of Functions
Example
The graph of f (x) =x2 − 2x− 3
x + 1:
f (x) =x2 − 2x− 3
x + 1=
(x− 3)(x + 1)(x + 1)
= x− 3 if x 6= 1
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
Math 53 (Part 1) Review of Functions 8 November 2012 13 / 69
![Page 40: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/40.jpg)
Graphs of Functions
Example
The graph of f (x) =x2 − 2x− 3
x + 1:
f (x) =x2 − 2x− 3
x + 1=
(x− 3)(x + 1)(x + 1)
= x− 3 if x 6= 1
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
Math 53 (Part 1) Review of Functions 8 November 2012 13 / 69
![Page 41: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/41.jpg)
Graphs of Functions
Example
The graph of f (x) =x2 − 2x− 3
x + 1:
f (x) =x2 − 2x− 3
x + 1=
(x− 3)(x + 1)(x + 1)
= x− 3 if x 6= 1
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
Math 53 (Part 1) Review of Functions 8 November 2012 13 / 69
![Page 42: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/42.jpg)
Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
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![Page 43: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/43.jpg)
Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))
Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
Math 53 (Part 1) Review of Functions 8 November 2012 14 / 69
![Page 44: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/44.jpg)
Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
Math 53 (Part 1) Review of Functions 8 November 2012 14 / 69
![Page 45: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/45.jpg)
Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
Math 53 (Part 1) Review of Functions 8 November 2012 14 / 69
![Page 46: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/46.jpg)
Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
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Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
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Graphs of Functions
Graphically:
Coordinates of a point on the graph in terms of x: (x, f (x))Domain: x-interval covered by the graph
Range: y-interval covered by the graph
Zero of a function: x-intercept of the graph
Intervals where the function value (or y-value) is positive: portions where thegraph lies above the x-axis
Intervals where the function value (or y-value) is negative: portions where thegraph lies below the x-axis
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Consider the graph of f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
(x, x− 3)
Domain:
R \ {−1}
Range:
R \ {−4}
Zero:
x = 3
Positive:
(3,+∞)
Negative:
(−∞,−1) ∪ (−1, 3)
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Consider the graph of f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
(x, x− 3)
Domain: R \ {−1}Range:
R \ {−4}
Zero:
x = 3
Positive:
(3,+∞)
Negative:
(−∞,−1) ∪ (−1, 3)
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Consider the graph of f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
(x, x− 3)
Domain: R \ {−1}Range: R \ {−4}Zero:
x = 3
Positive:
(3,+∞)
Negative:
(−∞,−1) ∪ (−1, 3)
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Consider the graph of f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
(x, x− 3)
Domain: R \ {−1}Range: R \ {−4}Zero: x = 3
Positive:
(3,+∞)
Negative:
(−∞,−1) ∪ (−1, 3)
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![Page 53: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/53.jpg)
Consider the graph of f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
(x, x− 3)
Domain: R \ {−1}Range: R \ {−4}Zero: x = 3
Positive: (3,+∞)Negative:
(−∞,−1) ∪ (−1, 3)
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Consider the graph of f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
(x, x− 3)
Domain: R \ {−1}Range: R \ {−4}Zero: x = 3
Positive: (3,+∞)Negative: (−∞,−1) ∪ (−1, 3)
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Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
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Basic types of functions
Constant Functions - functions of the form f (x) = c, where c is a real number
dom f = R; ran f = {c}graph: horizontal line intersecting the y-axis at y = c
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Basic types of functions
Constant Functions - functions of the form f (x) = c, where c is a real number
dom f = R; ran f = {c}graph: horizontal line intersecting the y-axis at y = c
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Basic types of functions
ExampleConsider the constant function f (x) = 2.
−3 −2 −1 1 2 3
−1
1
2
3
0
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Basic types of functions
Linear Functions - functions of the form
f (x) = mx + b
with m 6= 0
dom f = R; ran f = R
graph: m is slope; y-intercept is b
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Basic types of functions
Linear Functions - functions of the form
f (x) = mx + b
with m 6= 0
dom f = R; ran f = R
graph: m is slope; y-intercept is b
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Basic types of functions
ExampleConsider the linear function f (x) = −x + 1.
−2 −1 1 2
−1
1
2
3
0
m = −1, y-intercept: 1
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward, ran f =
[4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward, ran f =
[4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at
(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward, ran f =
[4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)
If a > 0: parabola opens upward, ran f =[
4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0:
parabola opens upward, ran f =[
4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward,
ran f =[
4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward, ran f =
[4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward, ran f =
[4ac−b2
4a ,+∞)
If a < 0:
parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Quadratic Functions - functions of the form
f (x) = ax2 + bx + c
with a 6= 0
dom f = R
graph: parabola with vertex at(− b
2a , 4ac−b2
4a
)If a > 0: parabola opens upward, ran f =
[4ac−b2
4a ,+∞)
If a < 0: parabola opens downward, ran f =(−∞, 4ac−b2
4a
]
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Basic types of functions
Example
Consider the quadratic function f (x) = x2.
−2 −1 1 2
1
2
3
4
0
A parabola opening upward with vertex at (0, 0)
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Basic types of functions
Example
Consider the quadratic function f (x) = −x2 − 2x + 3.
−3 −2 −1 1 2
−2
−1
1
2
3
4
0
A parabola opening downward with vertex at (−1, 4)
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Basic types of functions
Extreme function values of a quadratic function:
a > 0: f has a minimum function valuea < 0: f has a maximum function value
The extreme function value of f occurs at x = − b2a and the extreme function
value of f is f(− b
2a
)= 4ac−b2
4a
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Basic types of functions
Example
Consider the quadratic function f (x) = −x2 − 2x + 3.
Since a < 0
f has a maximum function value which occurs at x = −1The maximum value of f (x) is f (−1) = 4
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Basic types of functions
Example
Consider the quadratic function f (x) = −x2 − 2x + 3.
Since a < 0
f has a maximum function value which occurs at x = −1The maximum value of f (x) is f (−1) = 4
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Basic types of functions
Example
Consider the quadratic function f (x) = −x2 − 2x + 3.
Since a < 0
f has a maximum function value which occurs at x = −1The maximum value of f (x) is f (−1) = 4
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Basic types of functions
Polynomial Functions - functions of the form
f (x) = anxn + an−1xn−1 + · · ·+ a1x + a0
where n ∈W, an, an−1, ..., a0 are real numbers, with an 6= 0.
leading coefficient: an
degree of f : ndom f = R
Constant, linear and quadratic functions are special types of polynomialfunctions
Graphs of polynomial functions: Unit 3
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Basic types of functions
Polynomial Functions - functions of the form
f (x) = anxn + an−1xn−1 + · · ·+ a1x + a0
where n ∈W, an, an−1, ..., a0 are real numbers, with an 6= 0.
leading coefficient: an
degree of f : n
dom f = R
Constant, linear and quadratic functions are special types of polynomialfunctions
Graphs of polynomial functions: Unit 3
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Basic types of functions
Polynomial Functions - functions of the form
f (x) = anxn + an−1xn−1 + · · ·+ a1x + a0
where n ∈W, an, an−1, ..., a0 are real numbers, with an 6= 0.
leading coefficient: an
degree of f : ndom f = R
Constant, linear and quadratic functions are special types of polynomialfunctions
Graphs of polynomial functions: Unit 3
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Basic types of functions
Polynomial Functions - functions of the form
f (x) = anxn + an−1xn−1 + · · ·+ a1x + a0
where n ∈W, an, an−1, ..., a0 are real numbers, with an 6= 0.
leading coefficient: an
degree of f : ndom f = R
Constant, linear and quadratic functions are special types of polynomialfunctions
Graphs of polynomial functions: Unit 3
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Basic types of functions
Polynomial Functions - functions of the form
f (x) = anxn + an−1xn−1 + · · ·+ a1x + a0
where n ∈W, an, an−1, ..., a0 are real numbers, with an 6= 0.
leading coefficient: an
degree of f : ndom f = R
Constant, linear and quadratic functions are special types of polynomialfunctions
Graphs of polynomial functions: Unit 3
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Basic types of functions
Rational Functions - functions of the form f (x) =p(x)q(x)
, where p and q are
polynomial functions, and q is not the constant zero function.
Domain: {x ∈ R | q(x) 6= 0}Graphs of rational functions: Unit 3
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Basic types of functions
Rational Functions - functions of the form f (x) =p(x)q(x)
, where p and q are
polynomial functions, and q is not the constant zero function.
Domain: {x ∈ R | q(x) 6= 0}Graphs of rational functions: Unit 3
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Graphs of Functions
Example
Consider the rational function f (x) =x2 − 2x− 3
x + 1.
−3 −2 −1 1 2 3 4
−5
−4
−3
−2
−1
1
0
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Basic types of functions
Functions involving rational exponents or radicals - functions of the form
f (x) = n√
x = x1/n
n is odd: dom f = R
n is even: dom f = [0, ∞)
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Basic types of functions
Functions involving rational exponents or radicals - functions of the form
f (x) = n√
x = x1/n
n is odd: dom f = R
n is even: dom f = [0, ∞)
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Basic types of functions
ExampleSquare root function: f (x) =
√x
y =√
x =⇒ y2 = x, y ≥ 0
1 2 3 4
−2
−1
1
2
0
The graph of x = y2
1 2 3 4
−2
−1
1
2
0
The graph of y =√
x
The graph of f (x) =√
x is the upper branch of the parabola x = y2
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Basic types of functions
ExampleSquare root function: f (x) =
√x
y =√
x =⇒ y2 = x, y ≥ 0
1 2 3 4
−2
−1
1
2
0
The graph of x = y2
1 2 3 4
−2
−1
1
2
0
The graph of y =√
x
The graph of f (x) =√
x is the upper branch of the parabola x = y2
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Basic types of functions
ExampleSquare root function: f (x) =
√x
y =√
x =⇒ y2 = x, y ≥ 0
1 2 3 4
−2
−1
1
2
0
The graph of x = y2
1 2 3 4
−2
−1
1
2
0
The graph of y =√
x
The graph of f (x) =√
x is the upper branch of the parabola x = y2
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Basic types of functions
Trigonometric/Circular Functions
sine, cosine, tangent, cotangent, secant and cosecant functions
In Math 53, the trigonometric functions are viewed as functions on the set ofreal numbers.
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Basic types of functions
Trigonometric/Circular Functions
sine, cosine, tangent, cotangent, secant and cosecant functions
In Math 53, the trigonometric functions are viewed as functions on the set ofreal numbers.
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Basic types of functions
Examplef (x) = sin x
−π−2π π 2π 3π 4π− π2− 3π
2π2
3π2
5π2
7π2
−1
1
0
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Basic types of functions
Examplef (x) = cos x
−π−2π π 2π 3π 4π− π2− 3π
2π2
3π2
5π2
7π2
−1
1
0
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Basic Types of Functions
Examplef (x) = tan x
−π π 2π− π2− 3π
2π2
3π2
5π2
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Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
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Constructing a table of signs
The table of signs show when a given mathematical expression is positive, zero ornegative.
Two Methods:
1 Interval Method
2 Test Value Method
In both cases, one must determine the numbers where the given mathematicalexpression is zero or undefined.
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Constructing a table of signs
The table of signs show when a given mathematical expression is positive, zero ornegative.
Two Methods:
1 Interval Method
2 Test Value Method
In both cases, one must determine the numbers where the given mathematicalexpression is zero or undefined.
Math 53 (Part 1) Review of Functions 8 November 2012 36 / 69
![Page 98: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/98.jpg)
Constructing a table of signs
The table of signs show when a given mathematical expression is positive, zero ornegative.
Two Methods:
1 Interval Method
2 Test Value Method
In both cases, one must determine the numbers where the given mathematicalexpression is zero or undefined.
Math 53 (Part 1) Review of Functions 8 November 2012 36 / 69
![Page 99: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/99.jpg)
Constructing a table of signs
Example
Determine the intervals where the the graph of f (x) =2x2 − x3
2x2 − 3x + 1lies above
the x-axis.
We want to determine the intervals for which
2x2 − x3
2x2 − 3x + 1> 0
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Constructing a table of signs
Example
Determine the intervals where the the graph of f (x) =2x2 − x3
2x2 − 3x + 1lies above
the x-axis.
We want to determine the intervals for which
2x2 − x3
2x2 − 3x + 1
> 0
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Constructing a table of signs
Example
Determine the intervals where the the graph of f (x) =2x2 − x3
2x2 − 3x + 1lies above
the x-axis.
We want to determine the intervals for which
2x2 − x3
2x2 − 3x + 1> 0
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1
=x2(2− x)
(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2
+ + + + +
2x− 1
− − + + +
x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2
+ + + + +
2x− 1
− − + + +
x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2
+ + + + +
2x− 1
− − + + +
x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2
+ + + + +
2x− 1
− − + + +
x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1
− − + + +
x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − −
+ + +
x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1
− − − + +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − −
+ +
2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x
+ + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + +
−
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+
+ − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+ +
− + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+ + −
+ −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − +
−
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 117: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/117.jpg)
Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 118: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/118.jpg)
Constructing a table of signsInterval Method: rewrite the expression as a product of factors whose table ofsigns are easily determined.
2x2 − x3
2x2 − 3x + 1=
x2(2− x)(2x− 1)(x− 1)
Zero at: x = 0, 2, Undefined at: x = 12 , 1
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2 + + + + +2x− 1 − − + + +x− 1 − − − + +2− x + + + + −
x2(2− x)(2x− 1)(x− 1)
+ + − + −
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 119: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/119.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 120: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/120.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + − + −
Sample point in (−∞, 0): x = −1
(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 121: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/121.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
Math 53 (Part 1) Review of Functions 8 November 2012 39 / 69
![Page 122: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/122.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+
+ − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
Math 53 (Part 1) Review of Functions 8 November 2012 39 / 69
![Page 123: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/123.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+
+ − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):
(+)(+)
(−)(−)We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
Math 53 (Part 1) Review of Functions 8 November 2012 39 / 69
![Page 124: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/124.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+
+ − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)
We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ +
− + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)
We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + −
+ −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)
We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + − +
−
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)
We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 128: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/128.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)
We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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![Page 129: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/129.jpg)
Constructing a table of signs
Test Value Method: test a value in the specified interval
(−∞, 0)(
0, 12
) (12 , 1)
(1, 2) (2,+∞)
x2(2− x)(2x− 1)(x− 1)
+ + − + −
Sample point in (−∞, 0): x = −1(−1)2(3)(−3)(−2)
Sample point in(
0, 12
):(+)(+)
(−)(−)We get the same result:
The graph of f lies above the x-axis in the intervals (−∞, 0) ∪(
0, 12
)∪ (1, 2).
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ − + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ − + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ − + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1
+ − + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+
− + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ −
+ −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ − +
−
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ − + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Constructing a table of signs
Example
Find the domain of f (x) =√−5x
x2 − 1.
Domain: x ∈ R such that−5x
x2 − 1≥ 0
Zero at: x = 0, Undefined at: x = −1, 1
(−∞,−1) (−1, 0) (0, 1) (1,+∞)−5x
x2 − 1+ − + −
Therefore,dom f = (∞,−1) ∪ [0, 1)
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Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
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Piecewise-defined functions
Piecewise-defined functions are functions that are defined by more than oneexpression. Such functions can be written in the form
f (x) =
f1(x) if x ∈ X1f2(x) if x ∈ X2
... if...
fn(x) if x ∈ Xn
where X1, ..., Xn ⊆ R with Xi ∩ Xj = ∅ for all i 6= j.
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Piecewise-defined functions
ExampleAn example of a piecewise function is the signum function (or sign function),denoted by sgn and defined by
sgn x =
−1 if x < 00 if x = 01 if x > 0
−3 −2 −1 1 2 3
−1
1
0
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Piecewise-defined functions
ExampleAn example of a piecewise function is the signum function (or sign function),denoted by sgn and defined by
sgn x =
−1 if x < 00 if x = 01 if x > 0
−3 −2 −1 1 2 3
−1
1
0
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The Absolute Value Function
Absolute Value Function - denoted by |x| and defined by
|x| =√
x2 =
{x, x ≥ 0−x, x < 0
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The Absolute Value Function
The graph of f (x) = |x|
−3 −2 −1 1 2 3
1
2
3
0
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The Greatest Integer Function
Greatest Integer Function (GIF)
[[x]]: greatest integer less than or equal to x
Example
1 [[2.4]]
= 2
2 [[2]]
= 2
3 [[0]]
= 0
4 [[−2.1]]
= −3
5 [[−π]]
= −4
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The Greatest Integer Function
Greatest Integer Function (GIF)
[[x]]: greatest integer less than or equal to x
Example
1 [[2.4]] = 22 [[2]]
= 2
3 [[0]]
= 0
4 [[−2.1]]
= −3
5 [[−π]]
= −4
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The Greatest Integer Function
Greatest Integer Function (GIF)
[[x]]: greatest integer less than or equal to x
Example
1 [[2.4]] = 22 [[2]] = 23 [[0]]
= 0
4 [[−2.1]]
= −3
5 [[−π]]
= −4
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The Greatest Integer Function
Greatest Integer Function (GIF)
[[x]]: greatest integer less than or equal to x
Example
1 [[2.4]] = 22 [[2]] = 23 [[0]] = 04 [[−2.1]]
= −3
5 [[−π]]
= −4
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The Greatest Integer Function
Greatest Integer Function (GIF)
[[x]]: greatest integer less than or equal to x
Example
1 [[2.4]] = 22 [[2]] = 23 [[0]] = 04 [[−2.1]] = −35 [[−π]]
= −4
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The Greatest Integer Function
Greatest Integer Function (GIF)
[[x]]: greatest integer less than or equal to x
Example
1 [[2.4]] = 22 [[2]] = 23 [[0]] = 04 [[−2.1]] = −35 [[−π]] = −4
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1,
− 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x
< 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 0
0, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00,
0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x
< 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 1
1, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 2
2, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3
......
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3...
...
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3...
...
In general,
[[x]] = n,
for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3...
...
In general,
[[x]] = n, for
n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3...
...
In general,
[[x]] = n, for n ≤ x
< n + 1 where n ∈ Z
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The Greatest Integer Function
As a piecewise function:
[[x]] =
......
− 1, − 1 ≤ x < 00, 0 ≤ x < 11, 1 ≤ x < 22, 2 ≤ x < 3...
...
In general,
[[x]] = n, for n ≤ x < n + 1 where n ∈ Z
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The Greatest Integer FunctionThe graph of f (x) = [[x]]
−4 −3 −2 −1 1 2 3 4
−4
−3
−2
−1
1
2
3
4
0
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Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
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Operations on Functions
Definition (Operations on Functions)Let f and g be functions, c ∈ R.
1 Addition: ( f + g)(x) = f (x) + g(x); dom( f + g) = dom f ∩ dom g2 Subtraction: ( f − g)(x) = f (x)− g(x); dom( f − g) = dom f ∩ dom g3 Multiplication: ( f g)(x) = f (x)g(x); dom( f g) = dom f ∩ dom g
4 Division:(
fg
)(x) =
f (x)g(x)
;
dom(
fg
)= (dom f ∩ dom g) \ {x ∈ dom g | g(x) = 0}
5 Composition: ( f ◦ g)(x) = f (g(x));dom( f ◦ g) = {x ∈ dom g | g(x) ∈ dom f }
6 Scalar Multiplication: c f (x) = c ( f (x)); dom c f = dom f
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Operations on Functions
Example
Express the function F(x) = sin2 (3x− 1) as a composition of three functionslisted among the basic types of functions.
Let
f (x) = x2
g(x) = sin xh(x) = 3x− 1
ThenF(x) = ( f ◦ g ◦ h) (x)
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Operations on Functions
Example
Express the function F(x) = sin2 (3x− 1) as a composition of three functionslisted among the basic types of functions.
Let
f (x) = x2
g(x) = sin xh(x) = 3x− 1
ThenF(x) = ( f ◦ g ◦ h) (x)
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Operations on Functions
Example
Express the function F(x) = sin2 (3x− 1) as a composition of three functionslisted among the basic types of functions.
Let
f (x) = x2
g(x) = sin xh(x) = 3x− 1
ThenF(x) = ( f ◦ g ◦ h) (x)
Math 53 (Part 1) Review of Functions 8 November 2012 51 / 69
![Page 171: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/171.jpg)
Operations on Functions
Example
Express the function F(x) = sin2 (3x− 1) as a composition of three functionslisted among the basic types of functions.
Let
f (x) = x2
g(x) = sin x
h(x) = 3x− 1
ThenF(x) = ( f ◦ g ◦ h) (x)
Math 53 (Part 1) Review of Functions 8 November 2012 51 / 69
![Page 172: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/172.jpg)
Operations on Functions
Example
Express the function F(x) = sin2 (3x− 1) as a composition of three functionslisted among the basic types of functions.
Let
f (x) = x2
g(x) = sin xh(x) = 3x− 1
ThenF(x) = ( f ◦ g ◦ h) (x)
Math 53 (Part 1) Review of Functions 8 November 2012 51 / 69
![Page 173: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/173.jpg)
Operations on Functions
Example
Express the function F(x) = sin2 (3x− 1) as a composition of three functionslisted among the basic types of functions.
Let
f (x) = x2
g(x) = sin xh(x) = 3x− 1
ThenF(x) = ( f ◦ g ◦ h) (x)
Math 53 (Part 1) Review of Functions 8 November 2012 51 / 69
![Page 174: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/174.jpg)
Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
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Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)]
=1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
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Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
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Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
Math 53 (Part 1) Review of Functions 8 November 2012 52 / 69
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Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
Math 53 (Part 1) Review of Functions 8 November 2012 52 / 69
![Page 179: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/179.jpg)
Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
Math 53 (Part 1) Review of Functions 8 November 2012 52 / 69
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Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h
= 2x + h
Math 53 (Part 1) Review of Functions 8 November 2012 52 / 69
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Operations on Functions
Example
Let f (x) = x2 and g(x) = x + h. Find1h[( f ◦ g) (x)− f (x)].
1h[( f ◦ g) (x)− f (x)] =
1h[ f (g(x))− f (x)]
=f (x + h)− f (x)
h
=(x + h)2 − x2
h
=(x2 + 2xh + h2)− x2
h
=2xh + h2
h= 2x + h
Math 53 (Part 1) Review of Functions 8 November 2012 52 / 69
![Page 182: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/182.jpg)
Operations on Functions
Example
Let f (x) = |x| and g(x) = x2 − 1. Express ( f ◦ g) (x) as a piecewise function.
Recall:
|x| =√
x2 =
{x, x ≥ 0−x, x < 0
Therefore,
( f ◦ g) (x) = f (g(x)) = |x2 − 1| ={
x2 − 1, x2 − 1 ≥ 0−(x2 − 1), x2 − 1 < 0
Math 53 (Part 1) Review of Functions 8 November 2012 53 / 69
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Operations on Functions
Example
Let f (x) = |x| and g(x) = x2 − 1. Express ( f ◦ g) (x) as a piecewise function.
Recall:
|x| =√
x2 =
{x, x ≥ 0−x, x < 0
Therefore,
( f ◦ g) (x) = f (g(x)) = |x2 − 1| ={
x2 − 1, x2 − 1 ≥ 0−(x2 − 1), x2 − 1 < 0
Math 53 (Part 1) Review of Functions 8 November 2012 53 / 69
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Operations on Functions
Example
Let f (x) = |x| and g(x) = x2 − 1. Express ( f ◦ g) (x) as a piecewise function.
Recall:
|x| =√
x2 =
{x, x ≥ 0−x, x < 0
Therefore,
( f ◦ g) (x) = f (g(x)) = |x2 − 1| ={
x2 − 1, x2 − 1 ≥ 0−(x2 − 1), x2 − 1 < 0
Math 53 (Part 1) Review of Functions 8 November 2012 53 / 69
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Operations on Functions
Example
Let f (x) = |x| and g(x) = x2 − 1. Express ( f ◦ g) (x) as a piecewise function.
( f ◦ g) (x) = f (g(x)) = |x2 − 1| ={
x2 − 1, x2 − 1 ≥ 0−(x2 − 1), x2 − 1 < 0
(−∞,−1) (−1, 1) (1,+∞)
x− 1 − − +
x + 1 − + +
x2 − 1 + − +
( f ◦ g) (x) = |x2 − 1| ={
x2 − 1, x ≥ 1 or x ≤ 11− x2, −1 < x < 1
Math 53 (Part 1) Review of Functions 8 November 2012 54 / 69
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Operations on Functions
Example
Let f (x) = |x| and g(x) = x2 − 1. Express ( f ◦ g) (x) as a piecewise function.
( f ◦ g) (x) = f (g(x)) = |x2 − 1| ={
x2 − 1, x2 − 1 ≥ 0−(x2 − 1), x2 − 1 < 0
(−∞,−1) (−1, 1) (1,+∞)
x− 1 − − +
x + 1 − + +
x2 − 1 + − +
( f ◦ g) (x) = |x2 − 1| ={
x2 − 1, x ≥ 1 or x ≤ 11− x2, −1 < x < 1
Math 53 (Part 1) Review of Functions 8 November 2012 54 / 69
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Operations on Functions
Example
Let f (x) = |x| and g(x) = x2 − 1. Express ( f ◦ g) (x) as a piecewise function.
( f ◦ g) (x) = f (g(x)) = |x2 − 1| ={
x2 − 1, x2 − 1 ≥ 0−(x2 − 1), x2 − 1 < 0
(−∞,−1) (−1, 1) (1,+∞)
x− 1 − − +
x + 1 − + +
x2 − 1 + − +
( f ◦ g) (x) = |x2 − 1| ={
x2 − 1, x ≥ 1 or x ≤ 11− x2, −1 < x < 1
Math 53 (Part 1) Review of Functions 8 November 2012 54 / 69
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( f ◦ g) (x) = |x2 − 1| ={
x2 − 1, x ≥ 1 or x ≤ 11− x2, −1 < x < 1
−3 −2 −1 1 2 3
1
2
3
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n if n ≤ 2x + 1 < n + 1
n ≤ 2x + 1 < n + 1n− 1 ≤ 2x < nn− 1
2≤ x <
n2
Math 53 (Part 1) Review of Functions 8 November 2012 56 / 69
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n
if n ≤ 2x + 1 < n + 1
n ≤ 2x + 1 < n + 1n− 1 ≤ 2x < nn− 1
2≤ x <
n2
Math 53 (Part 1) Review of Functions 8 November 2012 56 / 69
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n if n ≤ 2x + 1 < n + 1
n ≤ 2x + 1 < n + 1n− 1 ≤ 2x < nn− 1
2≤ x <
n2
Math 53 (Part 1) Review of Functions 8 November 2012 56 / 69
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n if n ≤ 2x + 1 < n + 1
n ≤ 2x + 1 < n + 1
n− 1 ≤ 2x < nn− 1
2≤ x <
n2
Math 53 (Part 1) Review of Functions 8 November 2012 56 / 69
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n if n ≤ 2x + 1 < n + 1
n ≤ 2x + 1 < n + 1n− 1 ≤ 2x < n
n− 12
≤ x <n2
Math 53 (Part 1) Review of Functions 8 November 2012 56 / 69
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n if n ≤ 2x + 1 < n + 1
n ≤ 2x + 1 < n + 1n− 1 ≤ 2x < nn− 1
2≤ x <
n2
Math 53 (Part 1) Review of Functions 8 November 2012 56 / 69
![Page 195: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/195.jpg)
Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n, ifn− 1
2≤ x <
n2
[[2x + 1]] =
......
−1, if −1 ≤ x < − 12
0, if − 12 ≤ x < 0
1, if 0 ≤ x < 12
2, if 12 ≤ x < 1
......
Math 53 (Part 1) Review of Functions 8 November 2012 57 / 69
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Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n, ifn− 1
2≤ x <
n2
[[2x + 1]] =
......
−1, if −1 ≤ x < − 12
0, if − 12 ≤ x < 0
1, if 0 ≤ x < 12
2, if 12 ≤ x < 1
......
Math 53 (Part 1) Review of Functions 8 November 2012 57 / 69
![Page 197: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/197.jpg)
Operations on Functions
ExampleLet f (x) = 2x + 1 and g(x) = [[x]]. Express (g ◦ f ) (x) as a piecewise function.
(g ◦ f ) (x) = [[2x + 1]] = n, ifn− 1
2≤ x <
n2
[[2x + 1]] =
......
−1, if −1 ≤ x < − 12
0, if − 12 ≤ x < 0
1, if 0 ≤ x < 12
2, if 12 ≤ x < 1
......
Math 53 (Part 1) Review of Functions 8 November 2012 57 / 69
![Page 198: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/198.jpg)
−4 −3 −2 −1 1 2 3 4
−4
−3
−2
−1
1
2
3
4
0
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Other Graphing Examples
Example
Graph g(x) =√
2− x.
y =√
2− x
y2 = 2− x, y ≥ 0
x = 2− y2, y ≥ 0
The graph of g is the upper branch of the parabola x = 2− y2.
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Other Graphing Examples
Example
Graph g(x) =√
2− x.
y =√
2− x
y2 = 2− x, y ≥ 0
x = 2− y2, y ≥ 0
The graph of g is the upper branch of the parabola x = 2− y2.
Math 53 (Part 1) Review of Functions 8 November 2012 59 / 69
![Page 201: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/201.jpg)
Other Graphing Examples
Example
Graph g(x) =√
2− x.
y =√
2− x
y2 = 2− x, y ≥ 0
x = 2− y2, y ≥ 0
The graph of g is the upper branch of the parabola x = 2− y2.
Math 53 (Part 1) Review of Functions 8 November 2012 59 / 69
![Page 202: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/202.jpg)
Other Graphing Examples
Example
Graph g(x) =√
2− x.
y =√
2− x
y2 = 2− x, y ≥ 0
x = 2− y2, y ≥ 0
The graph of g is the upper branch of the parabola x = 2− y2.
Math 53 (Part 1) Review of Functions 8 November 2012 59 / 69
![Page 203: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/203.jpg)
Other Graphing Examples
Example
Graph g(x) =√
2− x.
y =√
2− x
y2 = 2− x, y ≥ 0
x = 2− y2, y ≥ 0
The graph of g is the upper branch of the parabola x = 2− y2.
Math 53 (Part 1) Review of Functions 8 November 2012 59 / 69
![Page 204: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/204.jpg)
Other Graphing Examples
Example
Graph g(x) =√
2− x.
−2 −1 1 2
−2
−1
1
2
0
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Other Graphing Examples
Example
Graph h(x) = −√
4− x2.
y = −√
4− x2
y2 = 4− x2, y ≤ 0
x2 + y2 = 4, y ≤ 0
The graph of h is the lower semicircle of the circle x2 + y2 = 4.
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![Page 206: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/206.jpg)
Other Graphing Examples
Example
Graph h(x) = −√
4− x2.
y = −√
4− x2
y2 = 4− x2, y ≤ 0
x2 + y2 = 4, y ≤ 0
The graph of h is the lower semicircle of the circle x2 + y2 = 4.
Math 53 (Part 1) Review of Functions 8 November 2012 61 / 69
![Page 207: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/207.jpg)
Other Graphing Examples
Example
Graph h(x) = −√
4− x2.
y = −√
4− x2
y2 = 4− x2, y ≤ 0
x2 + y2 = 4, y ≤ 0
The graph of h is the lower semicircle of the circle x2 + y2 = 4.
Math 53 (Part 1) Review of Functions 8 November 2012 61 / 69
![Page 208: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/208.jpg)
Other Graphing Examples
Example
Graph h(x) = −√
4− x2.
y = −√
4− x2
y2 = 4− x2, y ≤ 0
x2 + y2 = 4, y ≤ 0
The graph of h is the lower semicircle of the circle x2 + y2 = 4.
Math 53 (Part 1) Review of Functions 8 November 2012 61 / 69
![Page 209: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/209.jpg)
Other Graphing Examples
Example
Graph h(x) = −√
4− x2.
y = −√
4− x2
y2 = 4− x2, y ≤ 0
x2 + y2 = 4, y ≤ 0
The graph of h is the lower semicircle of the circle x2 + y2 = 4.
Math 53 (Part 1) Review of Functions 8 November 2012 61 / 69
![Page 210: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/210.jpg)
Other Graphing Examples
Example
Graph h(x) = −√
4− x2.
−2 −1 1 2
−2
−1
1
2
Math 53 (Part 1) Review of Functions 8 November 2012 62 / 69
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Other Graphing Examples
Example
Graph f (x) =
x + 4 if x < −2
x3 + x2
x + 1if −2 ≤ x ≤ 2
|x− 6| if x > 2
.
f (x) =
x + 4 if x < −2
x2(x + 1)x + 1
if −2 ≤ x ≤ 2
−(x− 6) if 2 < x < 6x− 6 if x ≥ 6
f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
Math 53 (Part 1) Review of Functions 8 November 2012 63 / 69
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Other Graphing Examples
Example
Graph f (x) =
x + 4 if x < −2
x3 + x2
x + 1if −2 ≤ x ≤ 2
|x− 6| if x > 2
.
f (x) =
x + 4 if x < −2
x2(x + 1)x + 1
if −2 ≤ x ≤ 2
−(x− 6) if 2 < x < 6x− 6 if x ≥ 6
f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
Math 53 (Part 1) Review of Functions 8 November 2012 63 / 69
![Page 213: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/213.jpg)
Other Graphing Examples
Example
Graph f (x) =
x + 4 if x < −2
x3 + x2
x + 1if −2 ≤ x ≤ 2
|x− 6| if x > 2
.
f (x) =
x + 4 if x < −2
x2(x + 1)x + 1
if −2 ≤ x ≤ 2
−(x− 6) if 2 < x < 6x− 6 if x ≥ 6
f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
Math 53 (Part 1) Review of Functions 8 November 2012 63 / 69
![Page 214: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/214.jpg)
Other Graphing Examples
The graph of f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
−5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10−1
1
2
3
4
0
Math 53 (Part 1) Review of Functions 8 November 2012 64 / 69
![Page 215: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/215.jpg)
Other Graphing Examples
The graph of f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
−5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10−1
1
2
3
4
0
Math 53 (Part 1) Review of Functions 8 November 2012 64 / 69
![Page 216: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/216.jpg)
Other Graphing Examples
The graph of f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
−5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10−1
1
2
3
4
0
Math 53 (Part 1) Review of Functions 8 November 2012 64 / 69
![Page 217: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/217.jpg)
Other Graphing Examples
The graph of f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
−5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10−1
1
2
3
4
0
Math 53 (Part 1) Review of Functions 8 November 2012 64 / 69
![Page 218: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/218.jpg)
Other Graphing Examples
The graph of f (x) =
x + 4 if x < −2
x2 if −2 ≤ x ≤ 2, x 6= −1−x + 6 if 2 < x < 6x− 6 if x ≥ 6
−5 −4 −3 −2 −1 1 2 3 4 5 6 7 8 9 10−1
1
2
3
4
0
Math 53 (Part 1) Review of Functions 8 November 2012 64 / 69
![Page 219: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/219.jpg)
Outline
1 Functions
2 Basic Types of Functions
3 Constructing a table of signs
4 Piecewise-defined functions
5 Operations on Functions
6 Functions as Mathematical Models
Math 53 (Part 1) Review of Functions 8 November 2012 65 / 69
![Page 220: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/220.jpg)
Functions as Mathematical Models
Express a certain situation as a functional relationship between certainquantities
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![Page 221: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/221.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the fieldThe area A of the field:
A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 222: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/222.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the field
The area A of the field:A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 223: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/223.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the fieldThe area A of the field:
A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 224: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/224.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the fieldThe area A of the field:
A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 225: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/225.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the fieldThe area A of the field:
A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240
y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 226: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/226.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the fieldThe area A of the field:
A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 227: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/227.jpg)
Functions as Mathematical Models
ExampleA rectangular field has a perimeter of 240 meters. Express the area of the field asa function of its width.
Let x be the width and y be the length of the fieldThe area A of the field:
A = xy
Since the perimeter of the field is 240 meters:
2x + 2y = 240y = 120− x
The area of the field expressed as a function of x:
A(x) = x(120− x) = −x2 + 120x
Math 53 (Part 1) Review of Functions 8 November 2012 67 / 69
![Page 228: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/228.jpg)
Functions as Mathematical Models
ExampleFind two numbers whose difference is 14 and whose product is minimum.
Let x be the greater number and y be the smaller number.Since the difference of 14 is positive
x− y = 14y = x− 14
The product as a function of x is
P(x) = x(x− 14) = x2 − 14x
P is a quadratic function with a minimum function value at
x = − b2a
= 7
The two numbers are 7 and −7, and the minimum product is P(7) = −49.
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![Page 229: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/229.jpg)
Functions as Mathematical Models
ExampleFind two numbers whose difference is 14 and whose product is minimum.
Let x be the greater number and y be the smaller number.
Since the difference of 14 is positive
x− y = 14y = x− 14
The product as a function of x is
P(x) = x(x− 14) = x2 − 14x
P is a quadratic function with a minimum function value at
x = − b2a
= 7
The two numbers are 7 and −7, and the minimum product is P(7) = −49.
Math 53 (Part 1) Review of Functions 8 November 2012 68 / 69
![Page 230: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/230.jpg)
Functions as Mathematical Models
ExampleFind two numbers whose difference is 14 and whose product is minimum.
Let x be the greater number and y be the smaller number.Since the difference of 14 is positive
x− y = 14y = x− 14
The product as a function of x is
P(x) = x(x− 14) = x2 − 14x
P is a quadratic function with a minimum function value at
x = − b2a
= 7
The two numbers are 7 and −7, and the minimum product is P(7) = −49.
Math 53 (Part 1) Review of Functions 8 November 2012 68 / 69
![Page 231: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/231.jpg)
Functions as Mathematical Models
ExampleFind two numbers whose difference is 14 and whose product is minimum.
Let x be the greater number and y be the smaller number.Since the difference of 14 is positive
x− y = 14y = x− 14
The product as a function of x is
P(x) = x(x− 14) = x2 − 14x
P is a quadratic function with a minimum function value at
x = − b2a
= 7
The two numbers are 7 and −7, and the minimum product is P(7) = −49.
Math 53 (Part 1) Review of Functions 8 November 2012 68 / 69
![Page 232: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/232.jpg)
Functions as Mathematical Models
ExampleFind two numbers whose difference is 14 and whose product is minimum.
Let x be the greater number and y be the smaller number.Since the difference of 14 is positive
x− y = 14y = x− 14
The product as a function of x is
P(x) = x(x− 14) = x2 − 14x
P is a quadratic function with a minimum function value at
x = − b2a
= 7
The two numbers are 7 and −7, and the minimum product is P(7) = −49.
Math 53 (Part 1) Review of Functions 8 November 2012 68 / 69
![Page 233: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/233.jpg)
Functions as Mathematical Models
ExampleFind two numbers whose difference is 14 and whose product is minimum.
Let x be the greater number and y be the smaller number.Since the difference of 14 is positive
x− y = 14y = x− 14
The product as a function of x is
P(x) = x(x− 14) = x2 − 14x
P is a quadratic function with a minimum function value at
x = − b2a
= 7
The two numbers are 7 and −7, and the minimum product is P(7) = −49.Math 53 (Part 1) Review of Functions 8 November 2012 68 / 69
![Page 234: Lecture 1 - Review of Functions.pdf](https://reader031.vdocuments.site/reader031/viewer/2022013102/552dd7f05503466f768b47f1/html5/thumbnails/234.jpg)
Announcements
Unit 1 Module will be available on 16 November 2012.
Google site: https://sites.google.com/a/math.upd.edu.ph/m53-s2-1213
All lecture slides will be posted in the website.
A printer–friendly, condensed version of slides for the 1st three lectures willbe uploaded in the website prior to the lecture.
Math 53 (Part 1) Review of Functions 8 November 2012 69 / 69