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# Domain and Range

Module by: First Last. E-mail the author

Summary: In this section, you will:

• Find the domain of a function defined by an equation.
• Graph piecewise-defined functions.

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If you’re in the mood for a scary movie, you may want to check out one of the five most popular horror movies of all time—I am Legend, Hannibal, The Ring, The Grudge, and The Conjuring. Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

## Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation, we were introduced to the concepts of domain and range. In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See Figure 2.

### A General Note: Piecewise Function:

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

f(x)={ formula 1     if x is in domain 1 formula 2     if x is in domain 2 formula 3     if x is in domain 3 f(x)={ formula 1     if x is in domain 1 formula 2     if x is in domain 2 formula 3     if x is in domain 3

In piecewise notation, the absolute value function is

| x |={ x    if  x0 x  if  x<0 | x |={ x    if  x0 x  if  x<0

### How To:

Given a piecewise function, write the formula and identify the domain for each interval.

1. Identify the intervals for which different rules apply.
2. Determine formulas that describe how to calculate an output from an input in each interval.
3. Use braces and if-statements to write the function.

### Example 11

#### Problem 1

##### Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed$50 fee for a group of 10 or more people. Write a function relating the number of people, n, n,to the cost, C. C.

##### Solution

Two different formulas will be needed. For n-values under 10, C=5n. C=5n.For values of n nthat are 10 or greater, C=50. C=50.

C(n)={ 5n if 0<n<10 50 if n10 C(n)={ 5n if 0<n<10 50 if n10
##### Analysis

The function is represented in Figure 23. The graph is a diagonal line from n=0 n=0to n=10 n=10and a constant after that. In this example, the two formulas agree at the meeting point where n=10, n=10,but not all piecewise functions have this property.

### Example 12

#### Problem 1

##### Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, C, C,in dollars for g ggigabytes of data transfer.

C(g)={ 25 if 0<g<2 25+10(g2) if g2 C(g)={ 25 if 0<g<2 25+10(g2) if g2

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

##### Solution

To find the cost of using 1.5 gigabytes of data, C(1.5), C(1.5),we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

C(1.5)=$25 C(1.5)=$25

To find the cost of using 4 gigabytes of data, C(4), C(4),we see that our input of 4 is greater than 2, so we use the second formula.

C(4)=25+10(42)=$45 C(4)=25+10(42)=$45
##### Analysis

The function is represented in Figure 24. We can see where the function changes from a constant to a shifted and stretched identity at g=2. g=2.We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

### How To:

Given a piecewise function, sketch a graph.

1. Indicate on the x-axis the boundaries defined by the intervals on each piece of the domain.
2. For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

### Example 13

#### Problem 1

##### Graphing a Piecewise Function

Sketch a graph of the function.

f(x)={ x 2 if x1 3 if 1<x2 x if x>2 f(x)={ x 2 if x1 3 if 1<x2 x if x>2
##### Solution

Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Figure 25 shows the three components of the piecewise function graphed on separate coordinate systems.

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26.

##### Analysis

Note that the graph does pass the vertical line test even at x=1 x=1and x=2 x=2because the points (1,3)(1,3) and (2,2 )(2,2 ) are not part of the graph of the function, though (1,1)(1,1) and (2,3)(2,3) are.

### Try It:

#### Exercise 8

Graph the following piecewise function.

f(x)={ x 3 if x<1 2 if 1<x<4 x if x>4 f(x)={ x 3 if x<1 2 if 1<x<4 x if x>4

### Q&A:

Can more than one formula from a piecewise function be applied to a value in the domain?

No. Each value corresponds to one equation in a piecewise formula.

### Media:

Access these online resources for additional instruction and practice with domain and range.

## Key Concepts

• The domain of a function includes all real input values that would not cause us to attempt an undefined mathematical operation, such as dividing by zero or taking the square root of a negative number.
• The domain of a function can be determined by listing the input values of a set of ordered pairs. See Example 1.
• The domain of a function can also be determined by identifying the input values of a function written as an equation. See Example 2, Example 3, and Example 4.
• Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation. See Example 5.
• For many functions, the domain and range can be determined from a graph. See Example 6 and Example 7.
• An understanding of toolkit functions can be used to find the domain and range of related functions. See Example 8, Example 9, and Example 10.
• A piecewise function is described by more than one formula. See Example 11 and Example 12.
• A piecewise function can be graphed using each algebraic formula on its assigned subdomain. See Example 13.

## Section Exercises

### Verbal

#### Exercise 9

Why does the domain differ for different functions?

##### Solution

The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

#### Exercise 10

How do we determine the domain of a function defined by an equation?

#### Exercise 11

Explain why the domain of f(x)= x 3 f(x)= x 3 is different from the domain of f(x)= x . f(x)= x .

##### Solution

There is no restriction on x xfor f(x)= x 3 f(x)= x 3 because you can take the cube root of any real number. So the domain is all real numbers, (,). (,).When dealing with the set of real numbers, you cannot take the square root of negative numbers. So x x-values are restricted for f(x)= x f(x)= x to nonnegative numbers and the domain is [0,). [0,).

#### Exercise 12

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

#### Exercise 13

How do you graph a piecewise function?

##### Solution

Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the x x-axis and y y-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate or  .  . Combine the graphs to find the graph of the piecewise function.

### Algebraic

For the following exercises, find the domain of each function using interval notation.

#### Exercise 14

f(x)=2x(x1)(x2) f(x)=2x(x1)(x2)

#### Exercise 15

f(x)=52 x 2 f(x)=52 x 2

(,) (,)

#### Exercise 16

f( x )=3 x2 f( x )=3 x2

#### Exercise 17

f( x )=3 62x f( x )=3 62x

(,3] (,3]

#### Exercise 18

f(x)= 43x f(x)= 43x

#### Exercise 19

f(x)= x 2 +4 f(x)= x 2 +4

(,) (,)

#### Exercise 20

f(x)= 12x 3 f(x)= 12x 3

#### Exercise 21

f(x)= x1 3 f(x)= x1 3

(,) (,)

#### Exercise 22

f(x)= 9 x6 f(x)= 9 x6

#### Exercise 23

f( x )= 3x+1 4x+2 f( x )= 3x+1 4x+2

##### Solution

(, 1 2 )( 1 2 ,) (, 1 2 )( 1 2 ,)

#### Exercise 24

f( x )= x+4 x4 f( x )= x+4 x4

#### Exercise 25

f(x)= x3 x 2 +9x22 f(x)= x3 x 2 +9x22

##### Solution

(,11)(11,2)(2,) (,11)(11,2)(2,)

#### Exercise 26

f(x)= 1 x 2 x6 f(x)= 1 x 2 x6

#### Exercise 27

f(x)= 2 x 3 250 x 2 2x15 f(x)= 2 x 3 250 x 2 2x15

##### Solution

(,3)(3,5)(5,) (,3)(3,5)(5,)

5 x3 5 x3

2x+1 5x 2x+1 5x

(,5) (,5)

#### Exercise 30

f(x)= x4 x6 f(x)= x4 x6

#### Exercise 31

f(x)= x6 x4 f(x)= x6 x4

[6,) [6,)

#### Exercise 32

f(x)= x x f(x)= x x

#### Exercise 33

f(x)= x 2 9x x 2 81 f(x)= x 2 9x x 2 81

##### Solution

( ,9 )( 9,9 )( 9, ) ( ,9 )( 9,9 )( 9, )

#### Exercise 34

Find the domain of the function f(x)= 2 x 3 50x f(x)= 2 x 3 50x by:

1. using algebra.
2. graphing the function in the radicand and determining intervals on the x-axis for which the radicand is nonnegative.

### Graphical

For the following exercises, write the domain and range of each function using interval notation.

#### Exercise 35

##### Solution

domain: (2,8], (2,8],range [6,8) [6,8)

#### Exercise 37

##### Solution

domain: [4, 4], [4, 4],range: [0, 2] [0, 2]

#### Exercise 39

##### Solution

domain: [5, 3), [5, 3),range: [ 0,2 ] [ 0,2 ]

#### Exercise 41

##### Solution

domain: (,1], (,1],range: [0,) [0,)

#### Exercise 43

##### Solution

domain: [ 6, 1 6 ][ 1 6 ,6 ]; [ 6, 1 6 ][ 1 6 ,6 ];range: [ 6, 1 6 ][ 1 6 ,6 ] [ 6, 1 6 ][ 1 6 ,6 ]

#### Exercise 45

##### Solution

domain: [3, ); [3, );range: [0,) [0,)

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

#### Exercise 46

f(x)={ x+1 if x<2 2x3 if x2 f(x)={ x+1 if x<2 2x3 if x2

#### Exercise 47

f(x)={ 2x1 if x<1 1+x if x1 f(x)={ 2x1 if x<1 1+x if x1

domain: (,) (,)

#### Exercise 48

f(x)={ x+1ifx<0 x1ifx>0 f(x)={ x+1ifx<0 x1ifx>0

#### Exercise 49

f( x )={ 3 if x<0 x if x0 f( x )={ 3 if x<0 x if x0

domain: (,) (,)

#### Exercise 50

f(x)={ x 2       if x<0 1x  if x>0 f(x)={ x 2       if x<0 1x  if x>0

#### Exercise 51

f(x)={ x 2 x+2 ifx<0 ifx0 f(x)={ x 2 x+2 ifx<0 ifx0

domain: (,) (,)

#### Exercise 52

f( x )={ x+1 if x<1 x 3 if x1 f( x )={ x+1 if x<1 x 3 if x1

#### Exercise 53

f(x)={ |x| 1 ifx<2 ifx2 f(x)={ |x| 1 ifx<2 ifx2

domain: (,) (,)

### Numeric

For the following exercises, given each function f, f,evaluate f(−3),f(−2),f(−1), f(−3),f(−2),f(−1), and f(0). f(0).

#### Exercise 54

f(x)={ x+1 if x<2 2x3 if x2 f(x)={ x+1 if x<2 2x3 if x2

#### Exercise 55

f(x)={ 1 if x3 0 if x>3 f(x)={ 1 if x3 0 if x>3

##### Solution

f(3)=1; f(2)=0; f(1)=0; f(0)=0 f(3)=1; f(2)=0; f(1)=0; f(0)=0

#### Exercise 56

f(x)={ 2 x 2 +3 if x1 5x7 if x>1 f(x)={ 2 x 2 +3 if x1 5x7 if x>1

For the following exercises, given each function f, f,evaluate f(−1),f(0),f(2), f(−1),f(0),f(2),and f(4). f(4).

#### Exercise 57

f(x)={ 7x+3 if x<0 7x+6 if x0 f(x)={ 7x+3 if x<0 7x+6 if x0

##### Solution

f(1)=4; f(0)=6; f(2)=20; f(4)=34 f(1)=4; f(0)=6; f(2)=20; f(4)=34

#### Exercise 58

f( x )={ x 2 2 if x<2 4+| x5 | if x2 f( x )={ x 2 2 if x<2 4+| x5 | if x2

#### Exercise 59

f( x )={ 5x if x<0 3 if 0x3 x 2 if x>3 f( x )={ 5x if x<0 3 if 0x3 x 2 if x>3

##### Solution

f(1)=5; f(0)=3; f(2)=3; f(4)=16 f(1)=5; f(0)=3; f(2)=3; f(4)=16

For the following exercises, write the domain for the piecewise function in interval notation.

#### Exercise 60

f(x)={ x+1 ifx<2 2x3ifx2 f(x)={ x+1 ifx<2 2x3ifx2

#### Exercise 61

f(x)={ x 2 2 ifx<1 x 2 +2ifx>1 f(x)={ x 2 2 ifx<1 x 2 +2ifx>1

##### Solution

domain: (,1)(1,) (,1)(1,)

#### Exercise 62

f(x)={ 2x3 3 x 2 ifx<0 ifx2 f(x)={ 2x3 3 x 2 ifx<0 ifx2

### Technology

#### Exercise 63

Graph y= 1 x 2 y= 1 x 2 on the viewing window [−0.5,−0.1] [−0.5,−0.1]and [0.1,0.5]. [0.1,0.5].Determine the corresponding range for the viewing window. Show the graphs.

##### Solution

window: [0.5,0.1]; [0.5,0.1];range: [4, 100] [4, 100]

window: [0.1, 0.5]; [0.1, 0.5];range: [4, 100] [4, 100]

#### Exercise 64

Graph y= 1 x y= 1 x on the viewing window [−0.5,−0.1] [−0.5,−0.1]and [0.1, 0.5]. [0.1, 0.5].Determine the corresponding range for the viewing window. Show the graphs.

### Extension

#### Exercise 65

Suppose the range of a function f fis [−5, 8]. [−5, 8].What is the range of |f(x)|? |f(x)|?

[0, 8] [0, 8]

#### Exercise 66

Create a function in which the range is all nonnegative real numbers.

#### Exercise 67

Create a function in which the domain is x>2. x>2.

##### Solution

Many answers. One function is f(x)= 1 x2 . f(x)= 1 x2 .

### Real-World Applications

#### Exercise 68

The height h hof a projectile is a function of the time t tit is in the air. The height in feet for t tseconds is given by the function h(t)=−16 t 2 +96t. h(t)=−16 t 2 +96t. What is the domain of the function? What does the domain mean in the context of the problem?

##### Solution

The domain is [0, 6]; [0, 6];it takes 6 seconds for the projectile to leave the ground and return to the ground

#### Exercise 69

The cost in dollars of making x xitems is given by the function C(x)=10x+500. C(x)=10x+500.

1. The fixed cost is determined when zero items are produced. Find the fixed cost for this item.
2. What is the cost of making 25 items?
3. Suppose the maximum cost allowed is \$1500. What are the domain and range of the cost function, C(x)? C(x)?

## Footnotes

1. The Numbers: Where Data and the Movie Business Meet. “Box Office History for Horror Movies.” http://www.the-numbers.com/market/genre/Horror. Accessed 3/24/2014
2. http://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPAK2&f=A.

## Glossary

interval notation:
a method of describing a set that includes all numbers between a lower limit and an upper limit; the lower and upper values are listed between brackets or parentheses, a square bracket indicating inclusion in the set, and a parenthesis indicating exclusion
piecewise function:
a function in which more than one formula is used to define the output
set-builder notation:
a method of describing a set by a rule that all of its members obey; it takes the form {x|statement about x} {x|statement about x}

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