section 1.3 functions. what you should learn how to determine whether relations between two...
TRANSCRIPT
![Page 1: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/1.jpg)
Section 1.3 Functions
![Page 2: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/2.jpg)
What you should learn
• How to determine whether relations between two variables are functions
• How to use function notation and evaluate two functions
• How to find the domains of functions
• How to use functions to model and solve real-life problems
![Page 3: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/3.jpg)
Relation• A relation is a set of ordered pairs of real
numbers.
F = {(3, 2) (4, 1) (2, 4) (1, 3)}
• If I say (2, __ ) , can you fill in the blank?
G = {(3, 3) (4, 1) (2, 1) (1, 3)}
• If I say (4, __ ) , can you fill in the blank?
![Page 4: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/4.jpg)
• In a relation the set of all of the values of the independent variable is called the domain.
• What is the domain of F?
{3, 4, 2, 1}
• Does G = {(3, 3) (4, 1) (2, 1) (1, 3)} have the same domain?
DomainF = {(3, 2) (4, 1) (2, 4) (1, 3)}
![Page 5: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/5.jpg)
• In a relation the set of all of the values of the dependent variable is called the range.
• What is the range of G?
{3, 1}
• Does F = {(3, 2) (4, 1) (2, 4) (1, 3)} have the same range?
Range G = {(3, 3) (4, 1) (2, 1) (1, 3)}
![Page 6: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/6.jpg)
(Domain, Range)
• Notice the alphabetical characteristic of Domain and Range.
(x, y)
(a, b)
(abscissa, ordinate)
• Unfortunately (independent, dependent) breaks the rule.
![Page 7: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/7.jpg)
Function
• A function is a relation in which , for each value of the first component there is exactly one value of the second component.
H = {(3, 2) (4, 1) (3, 4) (1, 3)}
K = {(2, 3) (4, 1) (3, 2) (1, 3)}
• H is not a function,but K is a function.
![Page 8: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/8.jpg)
Definition of a Function (page 27)
• A function from set A to set B is a relation that assigns to each element x in the set A exactly one element y in the set B.
• The set A is the domain (or set of inputs) of the function f.
• The set B contains the range (or the set of outputs)
![Page 9: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/9.jpg)
Function Expressed as a Mapping
A
C
B
1
2
3
F =
{(A,1)
(C, 2)
(B, 3)}
Domain Range
![Page 10: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/10.jpg)
Since A goes to two ranges G is not a function.
Function Expressed as a Mapping
A
C
B
1
2
3
G =
{(A,1)
(C, 2)
(B, 3)
(A, 4)}
Domain Range
4
![Page 11: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/11.jpg)
Characteristics of a function from Set A to Set B (page 27)
1. Each element in A must be matched with an element in B.
2. Some elements in B may not be matched with any element in A. (leftovers)
3. Two or more elements in A may be matched with the same element in B.
4. An element in A (the domain) cannot be matched with two different elements in B.
![Page 12: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/12.jpg)
Four Ways to Represent a Function
1. Verbally by a sentence that describes how the input variable is related to the output variable.
2. Numerically by a table or a list of ordered pairs that matches input values with output values
3. Graphically by points on a graph in a coordinate plane in which the inputs are represented on the horizontal axis and the output values are represented by the vertical axis.
4. Algebraically by an equation in two variables.
![Page 13: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/13.jpg)
Testing for Functions Example 1a
Determine whether the relation represents y as a function of x.
The input value x is the number of representatives from a state, and the output value y is the number of senators.
(x, 2)• This is a constant function.
![Page 14: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/14.jpg)
Testing for Functions Example 1b
Determine whether the relation represents y as a function of x.
Since x = 2 has two outputs the table does not describe a function.
Input x Output y
2 11
2 10
3 8
4 5
5 1
![Page 15: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/15.jpg)
Testing for Functions Represented Algebraically Example 2a
• Solve for y
• For each value of x there is only one value for y.
• So y is a function of x.
12 yx
12 xy
![Page 16: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/16.jpg)
Testing for Functions Represented Algebraically Example 2b
• Solve for y
• For each value of x there are two values for y.
• So y is not a function of x.
12 yx
12 xy
1 xy
![Page 17: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/17.jpg)
Functional Notation
• y = F(x)
• F(x) read F of x
• It does not mean F × x (multiplication)
![Page 18: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/18.jpg)
Functional Notation
• Consider y = 2x + 5• Suppose that you wanted to tell someone
to substitute in x = 3 into an equation.
• With functional notation y = 2x + 5 becomes f(x) = 2x + 5.
• And f(3) means substitute in 3 everyplace you see an x.
![Page 19: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/19.jpg)
Example 3a Evaluating a FunctionFind g(2)
14)( 2 xxxg1)2(4)2()2( 2 g
184)2( g5)2( g
![Page 20: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/20.jpg)
Example 3b Evaluating a FunctionFind g(t)
14)( 2 xxxg1)(4)()( 2 tttg
14)( 2 tttg
![Page 21: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/21.jpg)
Example 3c Evaluating a FunctionFind g(x+2)
14)( 2 xxxg1)2(4)2()( 2 xxtg
184)44()( 2 xxxtg18444)( 2 xxxtg
5)( 2 xtg
![Page 22: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/22.jpg)
Example 4 A Piecewise-Defined Function
0,1
0,1)(
2
xx
xxxf
x y
-2 5
-1 2
0 -1
1 0
2 1
![Page 23: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/23.jpg)
The Domain of a Function
• The implied domain is the set of all real numbers for which the expression is defined.
• For what values of x is f(x) undefined?
4
1)(
2
xxf
2x
}2|{ xx
![Page 24: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/24.jpg)
The Domain of a Function
• The implied domain is the set of all real numbers for which the expression is defined.
• For what values of x is g(x) undefined?
8)( xxg 08 x
}8|{ xx
![Page 25: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/25.jpg)
Example 7 Baseball
• A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and an angle of 45°. The path of the ball is given by the function
• Will the baseball clear a10-foot fence located 300 feet from home plate?
30032.0)( 2 xxxf
![Page 26: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/26.jpg)
Example 9 Evaluating a Difference Quotient
For
find
74)( 2 xxxf
h
xfhxf )()(
![Page 27: Section 1.3 Functions. What you should learn How to determine whether relations between two variables are functions How to use function notation and evaluate](https://reader033.vdocuments.site/reader033/viewer/2022061518/56649eca5503460f94bd91b4/html5/thumbnails/27.jpg)
Homework
• Page 35
• 1 - 67 odd, 77, 79, 93