Section 8.1

Functions and Their

Representations

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Basic Concepts

• Representations of a Function

• Definition of a Function

• Identifying a Function

• Graphing Calculators (Optional)

The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.

Name

y = f(x)

Output Input

FUNCTION NOTATION

The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.

Representations of a Function

Verbal Representation (Words)

Numerical Representation (Table of Values)

Symbolic Representation (Formula)

Graphical Representation (Graph)

Diagrammatic Representation (Diagram)

Example

Evaluate each function f at the given value of x.a. f(x) = 2x – 4, x = –3 b.

Solutiona. b.

( ) ; 21

xf x x

x

( ) 2 4f x x

( 3) 2( 3) 4

( 3) 10

f

f

( ) ; 21

xf x x

x

2

(2)1 22

(2) 21

f

f

Example

Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3).Solutiona. Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars.

b. Numerical Representation x f(x)

$1.00 $0.06

$2.00 $0.12

$3.00 $0.18

$4.00 $0.24

Example (cont)

c. Symbolic Representation f(x) = 0.06xd. Graphical Representation

e. Diagrammatic Representation

X

Y

1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

0.6

0

1 ●

2 ●

3 ●

4 ●

● 0.06

● 0.12

● 0.18

● 0.24

f(3) = 0.18

Definition of a Function

A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair:

(x, y).

Input Output

A relation is a set of ordered pairs, and a function is a special type of relation.

A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value.

Function

The domain of f is the set of all x-values, and the range of f is the set of all y-values.

Example

Use the graph of f to find the function’s domain and range. SolutionThe arrows at the ends of the graph indicate that the graph extends indefinitely. Thus the domain includes all real numbers.The smallest y-value on the graph is y = −4. Thus the range is y ≥ −4.

X

Y

-5 -4 -3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

0

Example

Use f(x) to find the domain of f.

a. f(x) = 3x b.

Solutiona. Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers.

b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4.

1

4f x

x

Example

Determine whether the table of values represents a function.

x f(x)

2 −6

3 4

4 2

3 −1

1 0

Solution

The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.

If every vertical line intersects a graph at no more than one point, then the graph represents a function.

Vertical Line Test

Example

Determine whether the graphs represent functions.a. b.

Any vertical line will cross the graph at most once. Therefore the graph does represent a function.

The graph does not represent a function because there exist vertical lines that can intersect the graph twice.

Section 8.2

Linear Functions

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Basic Concepts

• Representations of Linear Functions

• Modeling Data with Linear Functions

• The Midpoint Formula (Optional)

, 0 and 1,xf x a a a

A function f defined by f(x) = mx + b, where m and b are constants, is a linear function.

LINEAR FUNCTION

Example

Determine whether f is a linear function. If f is a linear function, find values for m and b so that f(x) = mx + b.a. f(x) = 6 – 2x b. f(x) = 3x2 – 5Solutiona. Let m = –2 and b = 6. Then f(x) = −2x + 6, and f is a linear function.

b. Function f is not linear because its formula contains x2. The formula for a linear function cannot contain an x with an exponent other than 1.

Example

Use the table of values to determine whether f(x) could represent a linear function. If f could be linear, write the formula for f in the form f(x) = mx + b.

SolutionFor each unit increase in x, f(x) increases by 7 units so f(x) could be linear with a = 7. Because f(0) = 4, b = 4. thus f(x) = 7x + 4.

x 0 1 2 3

f(x) 4 11 18 25

Example

Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).SolutionBegin by creating a table.

x y

−1 −4

0 −3

1 −2

2 −1

Plot the points and sketch a line through the points.

Example (cont)

Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).

To evaluate f(4), first find x = 4 on the x-axis. Then find the corresponding y-value. Thus f(4) = 1.

The formula f(x) = ax + b may be interpreted as follows.

f(x) = mx + b

(New amount) = (Change) + (Fixed amount)When x represents time, change equals (rate of change) × (time).

f(x) = m × x + b(Future amount) = (Rate of change) × (Time) + (Initial amount)

MODELING DATA WITH A LINEAR FUNCTION

Example

Suppose that a moving truck costs $0.25 per mile and a fixed rental fee of $20. Find a formula for a linear function that models the rental fees.

SolutionTotal cost is found by multiplying $0.25 (rate per mile) by the number of miles driven x and then adding the fixed rental fee (fixed amount) of $20. Thus f(x) = 0.25x + 20.

Example

The temperature of a hot tub is recorded at regular intervals.

a. Discuss the temperature of the water during this time interval.

The temperature appears to be a constant 102°F.

b. Find a formula for a function f that models these data.Because the temperature is constant, the rate of

change is 0. Thus f(x) = 0x + 102 or f(x) = 102.

Elapsed Time (hours) 0 1 2 3

Temperature 102°F 102°F 102°F 102°F

Example (cont)

The temperature of a hot tub is recorded at regular intervals.

c. Sketch a graph of f together with the data.

Elapsed Time (hours) 0 1 2 3

Temperature 102°F 102°F 102°F 102°F

0

20

40

60

80

100

120

0 1 2 3

Time (hours)

Te

mp

era

ture

(d

eg

ree

s)

The midpoint of the line segment with endpoints(x1, y1) and (x2, y2) in the xy-plane is

Midpoint Formula in the xy-Plane (Optional)

1 2 1 2,2 2

x x y yM

Example

Find the midpoint of the line segment connecting the points (3, 4) and (5, 3).

Solution1 2 1 2,

2 2

x x y yM

3 5 4 3,

2 2

2 1,

2 2

11,

2

Section 8.3

Compound Inequalities

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Basic Concepts

• Symbolic Solutions and Number Lines

• Numerical and Graphical Solutions

• Interval Notation

Basic Concepts

A compound inequality consists of two inequalities joined by the words and or or.

2x > –5 and 2x ≤ 8

x + 3 ≥ 4 or x – 2 < –6

Example

Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4

Solutionx + 2 < 7 and 2x – 3 > 3

Substitute 4 into the given compound inequality.4 + 2 < 7 and 2(4) – 3 > 3 6 < 7 and 5 > 3 True and True

Both inequalities are true, so 4 is a solution.

Example (cont)

Determine whether the given x-values are solutions to the compound inequalities. x + 2 < 7 and 2x – 3 > 3 x = 4, –4

Solutionx + 2 < 7 and 2x – 3 > 3

Substitute –4 into the given compound inequality. –4 + 2 < 7 and 2(–4) – 3 > 3 – 2 < 7 and –11 > 3 True and False

To be a solution both inequalities must be true, so –4 is not a solution.

Symbolic Solutions and Number Lines

We can use a number line to graph solutions to compound inequalities, such as x < 7 and x > –3.

x < 7

x > –3

x < 7 and x > –3

Note: A bracket, either [ or ] or a closed circle is used when an inequality contains ≤ or ≥. A parenthesis, either ( or ), or an open circle is used when an inequality contains < or >.

Example

Solve 3x + 6 > 12 and 5 – x < 11 . Graph the solution.

Solution3x + 6 > 12 and 5 – x < 11

3 6x 6x

2x 6x

Example

Solve each inequality. Graph each solution set. Write the solution in set-builder notation. a. b. c.Solutiona. b.

6 2 10w 4 4 8y 4 2

53 3

w

6 2 2 22 10w 4 8w

6 2 10w

| 4 8w w

1 2y

4 4 8y

| 1 2 y y

Example (cont)

c.

4 25

3 3

w

4 2 5

3 3

w

4 23 5

33

33

w

4 2 15w

24 2 22 15w

6 13w

( 6) ( 11 ) 31 1w

6 13w 13 6w

| 13 6 w w

Example

Solve x + 3 < –2 or x + 3 > 2

Solutionx + 3 < –2 or x + 3 > 2 x < –5 or x > –1

Example

Write each expression in interval notation. a. –3 ≤ x < 7

b. x ≥ 4

c. x < –3 or x ≥ 5

d. {x|x > 0 and x ≤ 5}

e. {x|x ≤ 2 or x ≥ 5}

3,7

4,

, 3 5,

0,5

, 2 5,

Example

Solve 2x + 3 ≤ –3 or 2x + 3 ≥ 5

Solution

2x + 3 ≤ –3 or 2x + 3 ≥ 5

2x ≤ –6 or 2x ≥ 2

x ≤ –3 or x ≥ 1

The solution set may be written as (, 3] [1, )

Section 8.4

Other Functions and Their Properties

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Expressing Domain and Range in Interval Notation

• Absolute Value Function

• Polynomial Functions

• Rational Functions (Optional)

• Operations on Functions

Expressing Domain and Range in Interval Notation

The set of all valid inputs for a function is called the domain, and the set of all outputs from a function is called the range.

Rather than writing “the set of all real numbers” for the domain of f, we can use interval notation to express the domain as (−∞, ∞).

Example

Write the domain for each function in interval notation.

a. f(x) = 3x b.Solutiona. The expression 3x is defined for all real numbers x. Thus the domain of f is

b. The expression is defined except when x – 4 = 0 or x = 4. Thus the domain of f includes all real numbers except 4 and can be written

1

4f x

x

, .

, 4 4, .

Absolute Value Function

We can define the absolute value function by f(x) = |x|.

To graph y = |x|, we begin by making a table of values.

x |x|

–2 2

–1 1

0 0

1 1

2 2

Example

Sketch the graph of f(x) = |x – 3|. Write its domain and range in interval notation. SolutionStart by making a table of values.

x y

0 3

2 1

3 0

4 1

6 3

X

Y

-3 -2 -1 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

1

2

3

4

5

0

The domain of f is , .

The range of f is [0, ).

Polynomial Functions

The following expressions are examples of polynomials of one variable.

As a result, we say that the following are symbolic representations of polynomial functions of one variable.

2 3, , and 51 515 3x xx x

32( ) 3, , and ( 5 1 ( )) 1 5 5f g x x xx xx hx

Example

Determine whether f(x) represents a polynomial function. If possible, identify the type of polynomial function and its degree. a.

b.

c.

3( ) 6 2 7f x x x

3.5( ) 4f x x

4( )

5f x

x

cubic polynomial, of degree 3

not a polynomial function because the exponent on the variable is negative

not a polynomial

Example

A graph of is shown. Evaluate f(1) graphically and check your result symbolically.Solution

3( ) 5f x x x

To calculate f(–1) graphically find –1 on the x-axis and move down until the graph of f is reached. Then move horizontally to the y-axis.

f(1) = –4 3( 1) 5( ( )1 1)f 5 1

4

Example

Evaluate f(x) at the given value of x.

Solution

3 2( ) 4 3 7, 2f x x x x

3 2( ) 4 3 7, 2f x x x x

3 2( 2) 4( 2) 3( 2) 7f

( 2) 4( 8) 3(4) 7f

( 2) 32 12 7f

( 2) 27f

Example

Use and to evaluate each of the following.

Solution

2( ) 3 1f x x 2( ) 6g x x

a. ( )(1) b. ( )( 2) c. 0f

f g f gg

2a. ( ) 3 11 (1)

4

f

2( ) 61 ( )

5

1g

( )(1) (1) (1)

4 5

9

f g f g

Example (cont)

Use and to evaluate each of the following.

Solution

2( ) 3 1f x x 2( ) 6g x x

a. ( )(1) b. ( )( 2) c. 0f

f g f gg

2b. ( ) 3( ) 1

3(4

2

13

2

) 1

f 2( ) 6 (

6 4

2

2 )2g ( )( 2) ( 2) ( 2)

13 2

11

f g f g

Example (cont)

Use and to evaluate each of the following.

Solution

2( ) 3 1f x x 2( ) 6g x x

a. ( )(1) b. ( )( 2) c. 0f

f g f gg

c. 0f

g

0

00

ff

g g

2

2

3( ) 1 =

6 )

0

(0

1 =

6

Section 8.5

Absolute Value Equations and

Inequalities

Copyright © 2013, 2009, and 2005 Pearson Education, Inc.

Objectives

• Absolute Value Equations

• Absolute Value Inequalities

Absolute Value Equations

An equation that contains an absolute value is an absolute value equation.Examples: |x| = 2, |2x – 5| = 7, |6 – 2x| – 5 = 2

Consider the equation |x| = 2.This equation has two solutions: 2 and –2 because |2| = 2 and also |2| = 2.

Absolute Value Equations

Example

Solve each equation. a. |x| = 38 b. |x| = –4

Solutiona. |x| = 38

b. |x| = –4

38 and 38

no solutions

Example

Solve |2x – 7| = 5 symbolically.Solution2x – 7 = 5 or 2x – 7 = –5

The solutions are 1 and 6.

2x = 122x – 7 = 5 or 2x – 7 = –5

x = 6or 2x = 2 or x = 1

Numerical Solutionx 0 1 2 3 4 5 6

|2x –7| 7 5 3 1 1 3 5

Graphical Solution

Example

Solve. a. |6 – x| – 3 = 0 b.Solutiona. Start by adding 3 to each side.

1 7( 8)

2 8x

6 3x

or 6 63 3x x

3 or 9x x

3 or 9x x

The solutions are 3 and 9.

Example (cont)

b.

1 7( 8)

2 8x

1 7( 8)

2 8x

or 1 1

( 8) ( 8)2 2

7 7

8 8x x

4( 8) 7 or 4( 8) 7x x

Multiply by 8 to clear fractions.

4 32 7 or 4 32 7x x

4 39 or 4 25x x 39 25

or 4 4

x x

The solutions are 25/4 and 39/4.

Example

Solve.a. |3x – 2| = –7 b. |6 – 3x| = 0

Solutiona. |3x – 2| = –7 The absolute value is never negative, there are no solutions.

b. |6 – 3x| = 0

–3x = –6

x = 2

Example

Solve |3x| = |2x – 5|.

Solution3x = 2x – 5 or 3x = –(2x – 5) x = –5 or 3x = –2x + 5

The solutions are −5, and 1.

5x = 5

x = 1

Example

Solve each absolute value equation and inequality.a. |3 – 4x| = 5 b. |3 – 4x| < 5 c. |3 – 4x| > 5Solutiona. |3 – 4x| = 5

3 – 4x = 5 or 3 – 4x = –5 –4x = 2 or –4x = –8 x = –1/2 or x = 2

b. |3 – 4x| < 5 The solution includes x-values between, but not including –1/2 and 2. {x| –1/2 < x < 2} or (–1/2, 2)

Example (cont)

Solve each absolute value equation and inequality.a. |3 – 4x| = 5 b. |3 – 4x| < 5 c. |3 – 4x| > 5Solutionc. |3 – 4x| > 5

The solution includes x-values to the left of x = –1/2 or to the right of x = 2.

The solution set is: {x|x < –1/2 or x > 2}or (, –1/2) (2, )

Example

Solve Write the solution set in interval notation.

Solution

3 24.

4

x

3 24

4

x 3 2 3 2

4 or 44 4

x x

3 2 16 or 3 2 16x x

3 18 or 3 14x x

146 or

3x x

14 or 6

3x x 14

, 6,3

Example

An engineer is designing a circular cover for a container. The diameter d of the cover is to be 4.75 inches and must be accurate within 0.05 inch. Write an absolute value inequality that gives acceptable values for d.SolutionThe diameter d must satisfy 4.7 ≤ d ≤ 4.8.Subtracting 4.75 from each part gives –0.05 ≤ d – 4.75 ≤ 0.05, which is equivalent to |d – 4.75| ≤ 0.05.

Example

Solve if possible.a. b.

Solutiona.

Because the absolute value of an expression cannot be negative, |4 – 3x| is greater than 0 – 1 for every x-value. The solution set is all real numbers.

4 3 2 1x 6 5 0x

4 3 2 1x

4 3 1x

Example

Solve if possible.a. b.

Solutiona.

Because the absolute value is always greater than or equal to 0, no x-values satisfy this inequality. There are no solutions.

4 3 2 1x 6 5 0x

6 5 0x