College Algebra with Corequisite Support

# 3.2Domain and Range

College Algebra with Corequisite Support3.2 Domain and Range

### Learning Objectives

In this section, you will:

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

### Corequisite Skills

#### Learning Objectives

• Find the domain and range of a function (IA 3.5.1)

A relation is any set of ordered pairs, (x,y). A special type of relation, called a function, is studied extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.

When studying functions, it’s important to be able to identify potential input values, called the domain, and potential output values, called the range.

### Example 1

Find the domain of the following function: {(2, 10),(3, 10),(4, 20),(5, 30),(6, 40)}.

A graph of a function can always help in identifying domain and range. When graphing basic functions, we can scan the x-axis just as we read in English from left to right to help determine the domain. We will scan the y-axis from bottom to top to help determine the range. So, in finding both domain and range, we scan axes from smallest to largest to see which values are defined. Typically, we will use interval notation, where you show the endpoints of defined sets using parentheses (endpoint not included) or brackets (endpoint is included) to express both the domain, D, and range, R, of a relation or function.

### Example 2

Find the domain and range from graphs

Find the domain and range of the function f whose graph is shown below. When working with functions expressed as an equation, the following steps can help to identify the domain.

### How To

Given a function written in equation form, find the domain.
1. Step 1. Identify the input values.
2. Step 2. Identify any restrictions on the input and exclude those values from the domain.
3. Step 3. Write the domain in interval notation form, if possible.

### Example 3

Find the domain and range from equations

Find the domain of the function $f(x)=x2-3f(x)=x2-3$

#### Activity: Restrictions on the domain of functions

Without a calculator, complete the following:

$60=60=$ _____; $00=00=$ _____; $06=06=$ _____; $4=4=$ _____; $–4=–4=$ _____; $83=83=$ _____; $–83=–83=$ _____

Clearly describe the two “trouble spots” which prevent expressions from representing real numbers:

1. ________________________________________________________________________________
2. ________________________________________________________________________________
3. Keeping these trouble spots in mind, algebraically determine the domain of each function. Write each answer in interval notation below the function. Remember, looking at the graph of the function can always help in finding domain and range.
 $f(x)=x-5x+3f(x)=x-5x+3$ $g(x)=5x+7g(x)=5x+7$ $h(x)=-35x-17h(x)=-35x-17$ $f(x)=x2-9x2+7x+10f(x)=x2-9x2+7x+10$ D:________; D:________; D:________; D:________;
##### Practice Makes Perfect

Find the domain and range of a function.

1.

For the relation {(1,3),(2,6),(3,9),(4,12),(5,15)}:

Find the domain of the function, express using set notation.
Find the range of the function, express using set notation.

2.

Use the graph of the function to find its domain and range. Write the domain and range in interval notation. D: _________ R: _________

3.

Graph the following function below. Use this graph to help determine the domain and range and express using interval notation. $f(x)=-4x-3f(x)=-4x-3$

D: ________ R: _________

4.

Use the graph of the function to find its domain and range. Write the domain and range in interval notation. D: ________ R: _________

5.

Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=-2|x|+3f(x)=-2|x|+3$

D: ________ R: _________

6.

Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=x-3f(x)=x-3$

D: _________ R: __________

7.

Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=x+43f(x)=x+43$

D: _________ R: __________

8.

Graph the following function. Use this graph to help determine the domain and range and express using interval notation. $f(x)=(x-3)(2x+1)f(x)=(x-3)(2x+1)$

D: _________ R: __________

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.

Figure 1 Based on data compiled by www.the-numbers.com.3

### 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.

Figure 2

### 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:

In piecewise notation, the absolute value function is

### 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

#### 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, $C. C.$

#### Analysis

The function is represented in Figure 23. The graph is a diagonal line from $n=0 n=0$ to $n=10 n=10$ and 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.

Figure 23

### Example 12

#### Working with a Piecewise Function

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

$C(g)={ 25 if 0

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

#### 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.

Figure 24

### 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

#### Graphing a Piecewise Function

Sketch a graph of the function.

$f(x)={ x 2 if x≤1 3 if 12 f(x)={ x 2 if x≤1 3 if 12$

#### Analysis

Note that the graph does pass the vertical line test even at $x=1 x=1$ and $x=2 x=2$ because 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 #8

Graph the following piecewise function.

$f(x)={ x 3 if x<−1 −2 if −14 f(x)={ x 3 if x<−1 −2 if −14$

### 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.

### 3.2 Section Exercises

#### Verbal

1.

Why does the domain differ for different functions?

2.

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

3.

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

4.

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

5.

How do you graph a piecewise function?

#### Algebraic

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

6.

$f(x)=−2x(x−1)(x−2) f(x)=−2x(x−1)(x−2)$

7.

$f(x)=5−2 x 2 f(x)=5−2 x 2$

8.

$f( x )=3 x−2 f( x )=3 x−2$

9.

$f( x )=3− 6−2x f( x )=3− 6−2x$

10.

$f(x)= 4−3x f(x)= 4−3x$

11.

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

12.

$f(x)= 1−2x 3 f(x)= 1−2x 3$

13.

$f(x)= x−1 3 f(x)= x−1 3$

14.

$f(x)= 9 x−6 f(x)= 9 x−6$

15.

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

16.

$f( x )= x+4 x−4 f( x )= x+4 x−4$

17.

$f(x)= x−3 x 2 +9x−22 f(x)= x−3 x 2 +9x−22$

18.

$f(x)= 1 x 2 −x−6 f(x)= 1 x 2 −x−6$

19.

$f(x)= 2 x 3 −250 x 2 −2x−15 f(x)= 2 x 3 −250 x 2 −2x−15$

20.

$5 x−3 5 x−3$

21.

$2x+1 5−x 2x+1 5−x$

22.

$f(x)= x−4 x−6 f(x)= x−4 x−6$

23.

$f(x)= x−6 x−4 f(x)= x−6 x−4$

24.

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

25.

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

26.

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.

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

38.

$f(x)={ x+1 if x<−2 −2x−3 if x≥−2 f(x)={ x+1 if x<−2 −2x−3 if x≥−2$

39.

$f(x)={ 2x−1 if x<1 1+x if x≥1 f(x)={ 2x−1 if x<1 1+x if x≥1$

40.

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

41.

$f( x )={ 3 if x<0 x if x≥0 f( x )={ 3 if x<0 x if x≥0$

42.

43.

$f(x)={ x 2 x+2 ifx<0 ifx≥0 f(x)={ x 2 x+2 ifx<0 ifx≥0$

44.

$f( x )={ x+1 if x<1 x 3 if x≥1 f( x )={ x+1 if x<1 x 3 if x≥1$

45.

$f(x)={ |x| 1 ifx<2 ifx≥2 f(x)={ |x| 1 ifx<2 ifx≥2$

#### Numeric

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

46.

$f(x)={ x+1 if x<−2 −2x−3 if x≥−2 f(x)={ x+1 if x<−2 −2x−3 if x≥−2$

47.

48.

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

49.

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

50.

$f( x )={ x 2 −2 if x<2 4+| x−5 | if x≥2 f( x )={ x 2 −2 if x<2 4+| x−5 | if x≥2$

51.

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

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

52.

$f(x)={ x+1ifx<−2 −2x−3ifx≥−2 f(x)={ x+1ifx<−2 −2x−3ifx≥−2$

53.

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

54.

$f(x)={ 2x−3 −3 x 2 ifx<0 ifx≥2 f(x)={ 2x−3 −3 x 2 ifx<0 ifx≥2$

#### Technology

55.

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.

56.

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

57.

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

58.

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

59.

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

#### Real-World Applications

60.

The height $h h$ of a projectile is a function of the time $t t$ it is in the air. The height in feet for $t t$ seconds 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?

61.

The cost in dollars of making $x x$ items 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

• 3The 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
• 4http://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPAK2&f=A.