Copyright © 2014, 2010, and 2006 Pearson Education, Inc. 1Copyright © 2014, 2010, and 2006 Pearson Education, Inc.

Chapter 7

Functions and Graphs

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Domain and Range

• Determining the Domain and the Range

• Restrictions on Domain

• Piecewise-Defined Functions

7.2

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FunctionA function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

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Example

Find the domain and range of the function f below.

6

54

2

3

-4

1

-2

-1

-3

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

f

-5

Here f can be written {(–5, 1), (1, 0), (3, –5), (4, 3)}.

The domain is the set of all first coordinates, {–5, 1, 3, 4}.

The range is the set of all second coordinates, {1, 0, –5, 3}.

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Example

For the function f represented below, determine each of the following.

a) What member of the range is paired with -2

b) The domain of f

c) What member of the domain is paired with 6

d) The range of f

y

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

f4

-2

-1

-4

-3

32

5

1

6

7

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a) What member of the range is paired with -2

SolutionLocate -2 on the horizontal axis (this is where the domain is located). Next, find the point directly above -2 on the graph of f. From that point, look to the corresponding y-coordinate, 3. The “input” -2 has the “output” 3. x

y

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

f4

-2

-1

-4

-3

32

5

1

6

Input

Output

7

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Solution

x

y

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

f4

-2

-1

-4

-3

32

5

1

6

The domain of f

7

b) The domain of f

The domain of f is the set of all x-values that are used in the points on the curve. These extend continuously from −5 to 3 and can be viewed as the curve’s shadow, or projection, on the x-axis. Thus the domain is { | 5 3}.x x

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c) What member of the domain is paired with 6

SolutionLocate 6 on the vertical axis (this is where the range is located). Next, find the point to the right of 6 on the graph of f. From that point, look to the corresponding x-coordinate, 2.5. The “output” 6 has the “input” 2.5.24

x

y

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

f4

-2

-1

-4

-3

32

5

1

6

Input

Output7

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d) The range of f

Solution

x

y

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

f4

-2

-1

-4

-3

32

5

1

6

The range of f

7

{ | 1 7}.y y

The range of f is the set of all y-values that are used in the points on the curve. These extend continuously from -1 to 7 and can be viewed as the curve’s shadow, or projection, on the y-axis. Thus the range is

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Example

Determine the domain of 2( ) 3 4.f x x

Solution

We ask, “Is there any number x for which we cannot compute 3x2 – 4?” Since the answer is no, the domain of f is the set of all real numbers.

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Example Determine the domain of

SolutionWe ask, “Is there any number x for which

cannot be computed?” Since cannot 2

8x be computed when x – 8 = 0 the answer is yes.

x – 8 = 0,x = 8

Thus 8 is not in the domain of f, whereas all other real numbers are. The domain of f is{ | is a real number 8}.x x and x

28x

2( ) .8

f xx

To determine what x-value would cause x – 8 to be 0, we solve an equation:

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Piecewise-Defined Functions

Some functions are defined by different equations for various parts of theirdomains. Such functions are said to be piecewise-defined. For example, the functiongiven by f(x) = |x| can be described by

To evaluate a piecewise-defined function for an input a, we determine what part of the domain a belongs to and use the appropriate formula for that part of the domain.

, if 0( )

, if 0x x

f xx x

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Example

Find each function value for the function f given by

a. f(5) b. f(–8) Solutiona. Determine which equation to use.

5 is in the second part of the domain

3 , if 4( )

2, if 4x x

f xx x

5 4( ) 2f x x

(5) 5 2 7f

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Example

Find each function value for the function f given by

a. f(5) b. f(–8) Solutionb. Determine which equation to use.

–8 is in the first part of the domain

3 , if 4( )

2, if 4x x

f xx x

8 4 ( ) 3f x x

( 8) 3( 8) 24f

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Example

Find each function value for the function f given by

a) f(3) b) f(2) c) f(7)Solutiona) f(x) = x + 3: f(3) = 3 + 3 = 0

b)f(x) = x2; f(2) = 22 = 4

c)f(7) = 4x = 4(7) = 28

2

3, if 3

( ) , if 3 44 , if 4

x x

f x x xx x