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TRANSCRIPT
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice HallCopyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Chapter 3
Graphs and Functions
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3.2
Introduction to Functions
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Relation, Domain, and Range
A relation is a set of ordered pairs.
The domain of the relation is the set of all first components of the ordered pairs.
The range of the relation is the set of all second components of the ordered pairs.
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Determine the domain and range of the relation
{(4,9), (–4,9), (2,3), (10, –5)}
Solution
The domain is the set of all the first coordinates of the ordered pairs.
Domain: 4, –4, 2, 10}.
The range is the set of all second coordinates of the ordered pairs.
Range: 9, 3, –5}.
Example 1a
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Input (Animal)
Polar Bear
Cow
Chimpanzee
Giraffe
Gorilla
Kangaroo
Red Fox
Output (Life Span)
20
15
10
7
Find the domain and range of the following relation.
Example 1b
Domain: {Polar Bear, Cow, Chimpanzee, Giraffe, Gorilla, Kangaroo, Red Fox}.
Range: {20, 15, 10, 7}.
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Some relations are also functions.
A function is a relation in which each first component in the ordered corresponds to exactly one second component.
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Given the relation {(4,9), (–4,9), (2,3), (10, –5)}, is it a function?
Solution
Since each element of the domain is paired with only one element of the range, it is a function.
Note: It is okay for a y-value to be assigned to more than one x-value, but an x-value cannot be assigned to more than one y-value (it has to be assigned to ONLY one y-value).
Example 2
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Is the relation y = x2 – 2x a function?
Solution
Each element of the domain (the x-values) would produce only one element of the range (the y-values), it is a function.
Example 3
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Is the relation x2 – y2 = 9 a function?
Solution
Each element of the domain (the x-values) would correspond with 2 different values of the range (both a positive and negative y-value), the relation is NOT a function.
Example 4
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Vertical Line TestIf no vertical line can be drawn so that it intersects a graph more than once, the graph is the graph of a function.
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Use the vertical line test to determine whether the graph to the right is the graph of a function.
x
y
Yes, this is the graph of a function since vertical line will intersect this graph more than once.
Example 5
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Use the vertical line test to determine whether the graph to the right is the graph of a function.
x
y
Example 5
Yes, this is the graph of a function since vertical line will intersect this graph more than once.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall
Use the vertical line test to determine whether the graph to the right is the graph of a function.
No, this is not the graph of a function. Vertical lines can be drawn that intersect the graph in two points.
x
y
Example 5
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Since the graph of a linear equation is a line, all linear equations are functions, except those whose graph is a vertical line
Note: An equation of the form y = c is a horizontal line and IS a function.
An equation of the form x = c is a vertical line and IS NOT a function.
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Find the domain and range of the function graphed to the right.
x
y
Domain: 3 ≤ x ≤ 4
Domain
Range: 4 ≤ y ≤ 2
Range
Example 6
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Find the domain and range of the function graphed to the right.
x
y
Domain: all real numbers
DomainRange: y ≥ – 2
Range
Example 6
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Function Notation
To denote that y is a function of x, we can write
y = f(x) (Read “f of x”)
Function notation
This notation means that y is a function of x or that y depends on x. For this reason, y is called the dependent variable and x the independent variable.
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If f(x) = x2 – 2x, find f(–3).
Solution
f(–3) = (–3)2 – 2(–3)
= 9 – (–6)
= 15
Example 7
f(x) = x2 – 2x
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Given the graph of the following function, find each function value by inspecting the graph.
f(5) = 6x
y
f(x)
f(4) = 3
f(5) = 1
f(6) = –6
Example 8