section 1.3 functions. what you should learn how to determine whether relations between two...

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 two functions

• How to find the domains of functions

• How to use functions to model and solve real-life problems

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?

• 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)}

• 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)}

(Domain, Range)

• Notice the alphabetical characteristic of Domain and Range.

(x, y)

(a, b)

(abscissa, ordinate)

• Unfortunately (independent, dependent) breaks the rule.

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.

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)

Function Expressed as a Mapping

A

C

B

1

2

3

F =

{(A,1)

(C, 2)

(B, 3)}

Domain Range

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

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.

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.

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.

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

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

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

Functional Notation

• y = F(x)

• F(x) read F of x

• It does not mean F × x (multiplication)

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.

Example 3a Evaluating a FunctionFind g(2)

14)( 2 xxxg1)2(4)2()2( 2 g

184)2( g5)2( g

Example 3b Evaluating a FunctionFind g(t)

14)( 2 xxxg1)(4)()( 2 tttg

14)( 2 tttg

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

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

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

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

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

Example 9 Evaluating a Difference Quotient

For

find

74)( 2 xxxf

h

xfhxf )()(

Homework

• Page 35

• 1 - 67 odd, 77, 79, 93