applying domain and range homework 6 answer key

Algebra Function Worksheets with Answer Keys

Feel free to download and enjoy these free worksheets on functions and relations .Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Domain and Range (Algebra 1)
• Functions vs Relations (distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations With Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)

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Maneuvering the Middle

Student-Centered Math Lessons

Teaching Domain and Range in Algebra 1

Domain and range is heavily emphasized in the Algebra 1 TEKS. The concept is found in two readiness standards and a supporting standard. Having taught it, I thought I would share some tips from my failures and successes. 

A.2(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real‐world situations , both continuous and discrete; and represent using inequalities  

Looking at former test questions helps me envision how standards can be tested. Here is an example from the 2017 Released STAAR test :

A.6(A) determine the domain and range of quadratic functions and represent it using inequalities

This test question example from the 2018 Release STAAR test : 

  Let’s jump into some ways that will help your students master domain and range!

1. Vocabulary is Key

This concept introduces new vocabulary that is necessary for using the skill. Make sure students have a solid understanding before moving forward. Don’t have them just copy it down. When was the last time you remembered something after writing it down one time? Encourage them to say it to a partner. Have students write a left to right arrow whenever they see the word domain and a vertical arrow whenever they see range. Make this sticky!  

I relied heavily on reading Math Equals Love when I taught Algebra 1 and 2. Sarah taught me to use the mnemonic tool DIXIROYD (which she credits to a unknown blog reader) originally to help students remember that D omain is the set of all I nput values, X values, and the I ndependent variable, while the R ange is the set of all O utput values, Y -values, and the D ependent variable. You know a mnemonic device is good when you rely on it just as much as your students.

Update 7/28/2026: Maneuvering the Middle now has a Middle School Math + Algebra 1 Word Wall.

As you can see in the video below, our Word Wall includes 190 essential math terms, their clear-cut definitions, and their visual representations.

We’ve included Spanish translations for all terms and definitions, ensuring a supportive and accessible learning experience for English Language Learners.

They were designed to be minimal prep and flexible to customize the formatting to suit your students’ unique needs.

2. Scaffold over a Few Lessons

As teachers, we are all guilty of jumping to an example that students are not ready for, which can cause them to become overwhelmed and shut down. Or worse, they develop false confidence and do an entire set of problems incorrectly.  I thought domain and range was really intuitive, so I tried to cover everything  in one class period and my students were lost. 

This skill requires scaffolding. I recommend starting with finding domain and range from tables and mappings first. Move to determining the domain and range from word problems or equations by creating an input and output table. Finally, move to graphs in this order — discrete graphs, graphs with endpoints, and then graphs with arrows. Interval notation can be tricky for students, so make sure to review  inequality symbols too.

3. Tools are Your Friend

I combed through the 2017 and 2018 STAAR tests and out of all the domain and range problems, 80% used a graph. Students have to be able to determine the domain and range by looking at a function on a graph. I recommend having students annotate the graph by use of colored pencils or highlighters. I have seen this done two ways. 

• Draw a box. Though I think this makes infinity less clear. 

Lastly, I used this foldable from Math Equals Love after my initial domain and range lesson bombed, and it was much more successful. 

Ready to teach it!? You can find our Properties of Functions Unit here

If you need hands-on Properties of Functions Activities, you can find them here (+Domain and Range)!

4. Use Technology for Real Time Application

We, like many teachers, are big fans of Desmos . A great lesson for introducing domain and range can be found here . For additional practice, I recommend this lesson .  Even if your students do not have access to technology , there is nothing stopping you from projecting these activities and leading a demonstration or discussion with your students. I firmly believe that the more students can manipulate a graph and see its effects, the more memorable or “sticky” that concept becomes!

UPDATE: ALGEBRA I DIGITAL ACTIVITIES ARE NOW AVAILABLE!

I love reading your comments about how you teach math. How do you teach domain and range in your classroom?

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March 12, 2020 at 12:32 pm

Is your domain and range worksheet from above available in your TPT store? I can’t find it.

March 27, 2020 at 1:23 pm

1.2 Domain and Range

Learning objectives.

In this section, you will:

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

Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s— 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 .

We can write the domain and range in interval notation , which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write ( 0 , 100 ] . ( 0 , 100 ] . We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The smallest term from the interval is written first.
• The largest term in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

See Figure 3 for a summary of interval notation.

Finding the Domain of a Function as a Set of Ordered Pairs

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

First identify the input values. The input value is the first coordinate in an ordered pair . There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

Find the domain of the function:

{ ( −5 , 4 ) , ( 0 , 0 ) , ( 5 , −4 ) , ( 10 , −8 ) , ( 15 , −12 ) } { ( −5 , 4 ) , ( 0 , 0 ) , ( 5 , −4 ) , ( 10 , −8 ) , ( 15 , −12 ) }

Given a function written in equation form, find the domain.

• Identify the input values.
• Identify any restrictions on the input and exclude those values from the domain.
• Write the domain in interval form, if possible.

Finding the Domain of a Function

Find the domain of the function f ( x ) = x 2 − 1. f ( x ) = x 2 − 1.

The input value, shown by the variable x x in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form, the domain of f f is ( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

Find the domain of the function: f ( x ) = 5 − x + x 3 . f ( x ) = 5 − x + x 3 .

Given a function written in an equation form that includes a fraction, find the domain.

• Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x x . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
• Write the domain in interval form, making sure to exclude any restricted values from the domain.

Finding the Domain of a Function Involving a Denominator

Find the domain of the function f ( x ) = x + 1 2 − x . f ( x ) = x + 1 2 − x .

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for x . x .

Now, we will exclude 2 from the domain. The answers are all real numbers where x < 2 x < 2 or x > 2. x > 2. We can use a symbol known as the union, ∪ , ∪ , to combine the two sets. In interval notation, we write the solution: ( −∞ , 2 ) ∪ ( 2 , ∞ ) . ( −∞ , 2 ) ∪ ( 2 , ∞ ) .

In interval form, the domain of f f is ( − ∞ , 2 ) ∪ ( 2 , ∞ ) . ( − ∞ , 2 ) ∪ ( 2 , ∞ ) .

Find the domain of the function: f ( x ) = 1 + 4 x 2 x − 1 . f ( x ) = 1 + 4 x 2 x − 1 .

Given a function written in equation form including an even root, find the domain.

• Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x . x .
• The solution(s) are the domain of the function. If possible, write the answer in interval form.

Finding the Domain of a Function with an Even Root

Find the domain of the function f ( x ) = 7 − x . f ( x ) = 7 − x .

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for x . x .

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7 , 7 , or ( − ∞ , 7 ] . ( − ∞ , 7 ] .

Find the domain of the function f ( x ) = 5 + 2 x . f ( x ) = 5 + 2 x .

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function f ( x ) = − 1 x f ( x ) = − 1 x has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation . For example, { x | 10 ≤ x < 30 } { x | 10 ≤ x < 30 } describes the behavior of x x in set-builder notation. The braces { } { } are read as “the set of,” and the vertical bar | is read as “such that,” so we would read { x | 10 ≤ x < 30 } { x | 10 ≤ x < 30 } as “the set of x -values such that 10 is less than or equal to x , x , and x x is less than 30.”

Figure 5 compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, ∪ , ∪ , to combine two unconnected intervals. For example, the union of the sets { 2 , 3 , 5 } { 2 , 3 , 5 } and { 4 , 6 } { 4 , 6 } is the set { 2 , 3 , 4 , 5 , 6 } . { 2 , 3 , 4 , 5 , 6 } . It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form { x | statement about  x } { x | statement about  x } which is read as, “the set of all x x such that the statement about x x is true.” For example,

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

Given a line graph, describe the set of values using interval notation.

• Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
• At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
• At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
• Use the union symbol ∪ ∪ to combine all intervals into one set.

Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.

To describe the values, x , x , included in the intervals shown, we would say, “ x x is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Given Figure 7 , specify the graphed set in

• ⓑ set-builder notation
• ⓒ interval notation

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x -axis. The range is the set of possible output values, which are shown on the y -axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8 .

We can observe that the graph extends horizontally from −5 −5 to the right without bound, so the domain is [ −5 , ∞ ) . [ −5 , ∞ ) . The vertical extent of the graph is all range values 5 5 and below, so the range is ( −∞ , 5 ] . ( −∞ , 5 ] . Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Finding Domain and Range from a Graph

Find the domain and range of the function f f whose graph is shown in Figure 9 .

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of f f is ( − 3 , 1 ] . ( − 3 , 1 ] .

The vertical extent of the graph is 0 to –4, so the range is [ − 4 , 0 ] . [ − 4 , 0 ] . See Figure 10 .

Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function f f whose graph is shown in Figure 11 .

The input quantity along the horizontal axis is “years,” which we represent with the variable t t for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable b b for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as 1973 ≤ t ≤ 2008 1973 ≤ t ≤ 2008 and the range as approximately 180 ≤ b ≤ 2010. 180 ≤ b ≤ 2010.

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Given Figure 12 , identify the domain and range using interval notation.

Can a function’s domain and range be the same?

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Given the formula for a function, determine the domain and range.

• Exclude from the domain any input values that result in division by zero.
• Exclude from the domain any input values that have nonreal (or undefined) number outputs.
• Use the valid input values to determine the range of the output values.
• Look at the function graph and table values to confirm the actual function behavior.

Finding the Domain and Range Using Toolkit Functions

Find the domain and range of f ( x ) = 2 x 3 − x . f ( x ) = 2 x 3 − x .

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is ( − ∞ , ∞ ) ( − ∞ , ∞ ) and the range is also ( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

Finding the Domain and Range

Find the domain and range of f ( x ) = 2 x + 1 . f ( x ) = 2 x + 1 .

We cannot evaluate the function at −1 −1 because division by zero is undefined. The domain is ( − ∞ , −1 ) ∪ ( −1 , ∞ ) . ( − ∞ , −1 ) ∪ ( −1 , ∞ ) . Because the function is never zero, we exclude 0 from the range. The range is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . ( − ∞ , 0 ) ∪ ( 0 , ∞ ) .

Find the domain and range of f ( x ) = 2 x + 4 . f ( x ) = 2 x + 4 .

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

The domain of f ( x ) f ( x ) is [ − 4 , ∞ ) . [ − 4 , ∞ ) .

We then find the range. We know that f ( − 4 ) = 0 , f ( − 4 ) = 0 , and the function value increases as x x increases without any upper limit. We conclude that the range of f f is [ 0 , ∞ ) . [ 0 , ∞ ) .

Figure 22 represents the function f . f .

Find the domain and range of f ( x ) = − 2 − x . f ( x ) = − 2 − x .

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function f ( x ) = | x | . f ( x ) = | x | . With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

If we input a negative value, the output is the opposite of the input.

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S S would be 0.1 S 0.1 S if S ≤ $ 10 , 000 S ≤ $ 10 , 000 and $ 1000 + 0.2 ( S − $ 10 , 000 ) $ 1000 + 0.2 ( S − $ 10 , 000 ) if S > $ 10 , 000. S > $ 10 , 000.

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

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

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

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 .

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

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.

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.

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

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.

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.

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.

Given a piecewise function, sketch a graph.

• Indicate on the x -axis the boundaries defined by the intervals on each piece of the domain.
• 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.

Graphing a Piecewise Function

Sketch a graph of the function.

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 .

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.

Graph the following piecewise function.

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.

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

• Domain and Range of Square Root Functions
• Determining Domain and Range
• Find Domain and Range Given the Graph
• Find Domain and Range Given a Table
• Find Domain and Range Given Points on a Coordinate Plane

1.2 Section Exercises

Why does the domain differ for different functions?

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

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 .

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

How do you graph a piecewise function?

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

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

f ( x ) = 5 − 2 x 2 f ( x ) = 5 − 2 x 2

f ( x ) = 3 x − 2 f ( x ) = 3 x − 2

f ( x ) = 3 − 6 − 2 x f ( x ) = 3 − 6 − 2 x

f ( x ) = 4 − 3 x f ( x ) = 4 − 3 x

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

f ( x ) = 1 − 2 x 3 f ( x ) = 1 − 2 x 3

f ( x ) = x − 1 3 f ( x ) = x − 1 3

f ( x ) = 9 x − 6 f ( x ) = 9 x − 6

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

f ( x ) = x + 4 x − 4 f ( x ) = x + 4 x − 4

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

f ( x ) = 1 x 2 − x − 6 f ( x ) = 1 x 2 − x − 6

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

f ( x ) = 5 x − 3 f ( x ) = 5 x − 3

f ( x ) = 2 x + 1 5 − x f ( x ) = 2 x + 1 5 − x

f ( x ) = x − 4 x − 6 f ( x ) = x − 4 x − 6

f ( x ) = x − 6 x − 4 f ( x ) = x − 6 x − 4

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

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

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

• ⓐ using algebra.
• ⓑ graphing the function in the radicand and determining intervals on the x -axis for which the radicand is nonnegative.

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

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

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

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

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

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

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

f ( x ) = { x 2 x + 2 if x < 0 if x ≥ 0 f ( x ) = { x 2 x + 2 if x < 0 if x ≥ 0

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

f ( x ) = { | x | 1 if x < 2 if x ≥ 2 f ( x ) = { | x | 1 if x < 2 if x ≥ 2

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 ) .

f ( x ) = { 1 if  x ≤ − 3 0 if  x > − 3 f ( x ) = { 1 if  x ≤ − 3 0 if  x > − 3

f ( x ) = { − 2 x 2 + 3 if  x ≤ − 1 5 x − 7 if  x > − 1 f ( x ) = { − 2 x 2 + 3 if  x ≤ − 1 5 x − 7 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 ) .

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

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

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

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

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

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

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.

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.

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

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

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

Real-World Applications

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 + 96 t . h ( t ) = −16 t 2 + 96 t . What is the domain of the function? What does the domain mean in the context of the problem?

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

• ⓐ The fixed cost is determined when zero items are produced. Find the fixed cost for this item.
• ⓑ What is the cost of making 25 items?
• ⓒ Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, C ( x ) ? C ( x ) ?
• 3 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
• 4 http://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPAK2&f=A.

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• Authors: Jay Abramson
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• Book title: Precalculus 2e
• Publication date: Dec 21, 2021
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• Book URL: https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
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3.3: Domain and Range

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Skills to Develop

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

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 \(\PageIndex{1}\) 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. For functions defined by an equation rather than by data, determining the domain and range requires a different kind of analysis.  In this section, we will investigate methods for determining the domain and range of functions such as these.

Figure \(\PageIndex{1}\): Graph of the Top-Five Grossing Horror Movies for years 2000-2003, and a Graph of the Market Share of Horror Movies by Year . Based on data compiled by www.the-numb ers.com.

In Section 1.1, 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. In this course, unless specifically stated otherwise, we are interested in functions whose inputs and outputs are always real numbers.  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 quotient, 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 outputs (Figure \(\PageIndex{2}\)).

Figure \(\PageIndex{2}\): Diagram of how a function relates two sets

Example \(\PageIndex{1}\): Finding the Domain of a Function as a Set of Ordered Pairs

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

First identify the input values. The input value is the first coordinate in an ordered pair. There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

\[D = \{2,3,4,5,6\} \nonumber\]

Next identify the output values.  The output value is the second coordinate in an ordered pair.  The range is the set of the second coordinates of the ordered pairs.  Do not list the same output twice.

\[R = \{10,20,30,40\}\nonumber \]

Find the domain and range of the function:

\[\{(−5,4),(0,0),(5,−4),(10,−8),(15,−12)\} \nonumber\]

\(D = \{−5, 0, 5, 10, 15\}\) \(R = \{4, 0, -4, -8, -12 \} \)

Finding the Domain of a Function Defined by an Equation

Let’s turn our attention to finding the domain of a function whose equation is provided. Finding the range is harder.  We will see later on in this section how to find the range if we have a graph of the function.  Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values from the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

We can often write the domain and range using interval notation, which uses values within brackets to describe a set of numbers (see Section 0.2). For example, if a person has $100 to spend, he or she might express the interval that is more than 0 and less than or equal to 100 and write \(\left(0, 100\right]\).   Note: This would not be quite correct, since this interval includes all real numbers between 0 and 100, and one cannot spend $\(\sqrt{5}\).  We can also use set-builder notation; for example, the interval \(\left(0, 100\right]\) can also be written as \(\{x \vert 0 < x \leq 100 \}\).

• Identify where the input values appear in the equation.
• Begin with the set of all real numbers as the possible domain.
• Identify any restrictions on the input and exclude those values from the set of real numbers.
• Write the domain in interval form or set-builder notation.

Example \(\PageIndex{2}\): Finding the Domain of a Function written in equation form

Find the domain of the function \(f(x)=x^2−1\).

The input value, shown by the variable \(x\) in the equation, is squared and then the result is decreased by one. Any real number may be squared and then be decreased by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form the domain of \(f\) is \(D = (−\infty,\infty)\), or alternatively \(D = \mathbb{R}\).

Find the domain of the function:

\[f(x)=5−x+x^3 \nonumber\]

\(D = (−\infty,\infty)\), or \(\mathbb{R}\)

• Identify any restrictions on the input. If the denominator contains an input value, set the denominator equal to zero and solve for \(x\) . The domain must be restricted to exclude all solutions to this equation.
• Write the domain in interval form, making sure to exclude any restricted values from the domain.

Example \(\PageIndex{3}\): Finding the Domain of a Function Involving a Denominator

Find the domain of the function \(f(x)=\dfrac{x+1}{2−x}\).

When there is a denominator, we want to exclude any \(x\)-values that force the denominator to be zero. To find these values, we set the denominator equal to 0 and solve for \(x\).

\[ \begin{align*} 2−x=0 \\[5pt] −x &=−2 \\[5pt] x&=2 \end{align*}\]

Now, we will exclude 2 from the domain. The domain consists of all real numbers that do not equal 2,  which means \(x<2\) or \(x>2\). We can use a symbol known as the union, \(\cup\), to combine the two sets.

In interval form, the domain of \(f\) is \(D = (−\infty,2)\cup(2,\infty)\).

\[f(x)=\dfrac{1+4x}{2x−1} \nonumber\]

\[ D = (−\infty,\dfrac{1}{2})\cup(\dfrac{1}{2},\infty) \nonumber\] Good for you, to try this problem! Get your professor to give you extra credit for Try It 1.2.3 .

• Identify any restrictions on the input.  Since there is an even root, only include inputs that do not result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for \(x\).
• The solution(s) are the domain of the function. If possible, write the answer in interval form.

Example \(\PageIndex{4}\): Finding the Domain of a Function with an Even Root

\[f(x)=\sqrt{7-x} \nonumber .\]

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for x.

\[ \begin{align*} 7−x&≥0 \\[5pt] −x&≥−7\\[5pt] x&≤7 \end{align*}\]

Now, we will include any real number less than or equal to 7 in the domain. The domain is \(D = \left(−\infty,7\right]\).

Find the domain of the function

\[f(x)=\sqrt{x+52}. \nonumber\]

\[\left[−52,\infty\right) \nonumber\]

Yes. For example, the function \(f(x)= -\sqrt{\frac{1}{x}}\) has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

Finding Domain and Range from Graphs

Another way to identify the domain of a function is by using graphs. A graph is also an excellent way to identify the range of a function.  Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the \(x\)-axis. The range is the set of possible output values, which are shown on the \(y\)-axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure \(\PageIndex{7}\).

Figure \(\PageIndex{7}\): Graph of a polynomial that shows the \(x\)-axis contains the domain and the \(y\)-axis contains the range

Observe that the graph extends horizontally from −5 to the right without bound, so the domain is \(\left[−5,∞\right)\). The vertical extent of the graph is all values from 5 and below, so the range is \(\left(−∞,5\right]\). Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Example \(\PageIndex{6A}\): Finding Domain and Range from a Graph

Find the domain and range of the function \(f\) whose graph is shown in Figure 1.2.8.

Figure \(\PageIndex{8}\): Graph of a function defined on (-3, 1].

Observe that the horizontal extent of the graph is –3 to 1, not including \(x=-3\) but including \(x=1\), so the domain of \(f\) is \(\left(−3,1\right]\).

The vertical extent of the graph is 0 to –4.  The range is \([−4,0]\).  There is no output for \(x=-3\) since \(-3\) is not part of the domain, so \(0\) is not the output for \(x=-3\); however, we can see that the point \((0,0)\) is on the graph, which means \(f(0)=0\), and so 0 is in the range.  See Figure \(\PageIndex{9}\).

Figure \(\PageIndex{9}\): Graph of the previous function showing the domain and range

Example \(\PageIndex{6B}\): Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function \(f\) whose graph is shown in Figure \(\PageIndex{10}\).

Figure \(\PageIndex{10}\): Graph of the Alaska Crude Oil Production where the vertical axis is thousand barrels per day and the horizontal axis is years (credit: modification of work by the U.S. Energy Information Administration)

The input quantity along the horizontal axis is “years,” which we represent with the variable \(t\) for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable \(b\) for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as \(1973≤t≤2008\) and the range as approximately \(180≤b≤2010\). Note: This is not really correct, because not every real number is included in the domain, nor in the range. The graph represents a mathematical model of a real-life situation.

In interval notation, the domain is \([1973, 2008]\), and the range is about \([180, 2010]\). For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Given Figure \(\PageIndex{11}\), identify the approximate domain and range using interval notation.

Figure \(\PageIndex{11}\): Graph of World Population Increase where the y-axis represents millions of people and the x-axis represents the year.

domain =\([1950,2000]\)

range = \([4.7\times 10^7,9.0\times 10^7]\)

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Figure \(\PageIndex{12}\): Constant function \(f(x)=c\).

For the constant function \( f(x)=c\), the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant \(c\), so the range is the set \(\{c\}\) that contains this single element. In interval notation, this could be written as \([c,c]\), the interval that both begins and ends with \(c\).

For the identity function \(f(x)=x\), there is no restriction on \(x\). Both the domain and range are the set of all real numbers.

Figure \(\PageIndex{14}\): Absolute function \(f(x)=|x|\).

For the absolute value function \(f(x)=|x|\), there is no restriction on \(x\). However, because absolute value is defined as a distance from 0, the output can only be greater than or equal to 0.

Figure \(\PageIndex{15}\): Quadratic function \(f(x)=x^2\).

For the quadratic function \(f(x)=x^2\), the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

Figure \(\PageIndex{16}\): Cubic function \(f(x)=x^3\).

For the cubic function \(f(x)=x^3\), the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.

Figure \(\PageIndex{17}\): Reciprocal function \(f(x)=\dfrac{1}{x}\).

For the reciprocal function \(f(x)=\dfrac{1}{x}\), we cannot divide by 0, so we must exclude 0 from the domain. Further, 1 divided by any value can never be 0, so the range also will not include 0. In set-builder notation, we could also write \(\{x| x≠0\}\), the set of all real numbers that are not zero, for both the domain and the range.

Figure \(\PageIndex{18}\): Reciprocal squared function \(f(x)=\dfrac{1}{x^2}\)

For the reciprocal squared function \(f(x)=\dfrac{1}{x^2}\),we cannot divide by 0, so we must exclude 0 from the domain. There is also no x that can give an output of 0, so 0 is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.

For the square root function \(f(x)=\sqrt{x}\), we cannot take the square root of a negative real number, so the domain must be 0 or greater. The range also excludes negative numbers because the square root symbol \(\sqrt{x}\) is defined to be the positive square root of \(x\), even though \(x\) also has a negative square root, denoted \(-\sqrt{x}\). 

Figure \(\PageIndex{20}\): Cube root function \(f(x)=\sqrt[3]{x}\) .

For the cube root function \(f(x)=\sqrt[3]{x}\), the domain and range include all real numbers. Note that there is no problem taking a cube root, or any odd-integer root, of a negative number, and the resulting output is negative (it is an odd function).

• Exclude from the domain any input values that result in division by zero.
• Exclude from the domain any input values that have nonreal (or undefined) number outputs.
• If possible, use the valid input values to determine the range of the output values.
• Look at the function graph and/or table values to confirm the actual function behavior.

Example \(\PageIndex{7}\) : Finding the Domain Using Toolkit Functions

Find the domain of \(f(x)=2x^3−x\).

This function uses two Toolkit Functions as building blocks. There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is \((−\infty,\infty)\).

Example \(\PageIndex{8}\): Finding the Domain

Find the domain of \(f(x)=\frac{2}{x+1}\).

We cannot evaluate the function at \(x=−1\) because division by zero is undefined. The domain is \((−\infty,−1)\cup(−1,\infty)\).

Example \(\PageIndex{9}\): Finding the Domain

Find the domain of \(f(x)=2 \sqrt{x+4}\).

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

\(x+4≥0\) when \(x≥−4\)

The domain of \(f(x)\) is \([−4,\infty)\).

We can also find the range, by comparing this function with the Toolkit Function \(g(x)=\sqrt{x}\). We know that \(f(−4)=0\), and the function value increases as \(x\) increases without any upper limit. We conclude that the range of \(f\) is \(\left[0,\infty\right)\).

Figure \(\PageIndex{19}\) represents the graph of the function \(f\).

Figure \(\PageIndex{19}\): Graph of a square root function which has been shifted to \((-4, 0)\).

Find the domain of

\(f(x)=\sqrt{−2−x}\).

domain: \(\left(−\infty,-2\right]\)

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function \(f(x)=|x|\). With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

Another way to define the absolute value function is to describe the outputs depending on where the inputs come from on the number line.  If we input 0, or a positive value, the output is the same as the input.

\[f(x)=x \mbox{ if }  x≥0 \nonumber \]

If we input a negative value, the output is the opposite of the input.  For example, \(f(-3) = -(-3) = 3\).

\[f(x)=−x \mbox{ if }  x<0 \nonumber\]

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function . A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

Piecewise Function

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

\[f(x)= \begin{cases} \text{formula 1} & \text{if x is in domain 1} \\ \text{formula 2} &\text{if x is in domain 2} \\ \text{formula 3} &\text{if x is in domain 3}\end{cases} \nonumber\]

In piecewise notation, the absolute value function is

\[|x|= \begin{cases} x & \text{if $x \geq 0$} \\ -x &\text{if $x<0$} \end{cases} \nonumber\]

The domain for the absolute value function is the union of \([0, \infty)\) and \((-\infty, 0)\); or, all real numbers.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” Tax brackets are a real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax \(T\) on a total income S would be

\[T(S)=\begin{cases}0.1S \mbox{ if } S≤$10,000 \\ \$1000+0.2(S−$10,000) \mbox{ if } S>$10,000.\end{cases} \nonumber\]

Can you give a clear explanation of the reason for the formula of each piece of this income tax function?  If so, write it up and give it to your instructor for Extra Credit .

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

Example \(\PageIndex{10}\): 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, \(n\), to the cost, \(C\).

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

\[C(n)= \begin{cases} 5n & \text{if $n < 10$} \\ 50 &\text{if $n\geq 10$} \end{cases} \nonumber\]

The function is represented as a graph in Figure \(\PageIndex{20}\). The graph is a straight line with slope of 5 from \(n=0\) to \(n=10\) and a constant after that. In this example, the two formulas agree at the meeting point where \(n=10\), but not all piecewise functions have this property.

Figure \(\PageIndex{20}\)

Note: Looking at the piecewise definition of function \(C\), it seems that the domain is \(D = (-\infty, \infty)\).  Looking at the graph, it seems that the domain for \(C\) is \(D=[0, \infty)\).  Given that this function is a mathematical model for a real-life situation, what do you think the domain is?  How should the definition of \(f\) be adjusted?  If you think you know, write an explanation and give it to your instructor for Extra Credit .

Example \(\PageIndex{11}\): Working with a Piecewise Function

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

\[C(g)= \begin{cases} 25 & \text{if $0<g<2$} \\ 25+10(g-2) &\text{if $g\geq2$} \end{cases} \nonumber\]

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

To find the cost of using 1.5 gigabytes of data, \(C(1.5)\), 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.

\[C(1.5)=$25 \nonumber\]

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

\[C(4)=25+10(4−2)=$45 \nonumber\]

The function is represented in Figure \(\PageIndex{21}\). We can see where the function changes from a constant to a linear function with positive slope at \(g=2\). 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 \(\PageIndex{21}\)

Note : Calculate the slope of the straight line defined as the second piece of the function \(C\), and give your work to your instructor for Extra Credit .

• Indicate on the x-axis the boundaries defined by the intervals on each piece of the domain.
• 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 \(\PageIndex{12}\): Graphing a Piecewise Function

Sketch a graph of the function.

\[f(x)= \begin{cases} x^2 & \text{if $x \leq 1$} \\ 3 &\text{if $1<x\leq2$} \\ x &\text{if $x>2$} \end{cases} \nonumber\]

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 \(\PageIndex{20}\) shows the three components of the piecewise function graphed on separate coordinate systems.

(a)\( f(x)=x^2\) if \(x≤1\); (b) \(f(x)=3\) if \(1< x≤2\); (c) \(f(x)=x\) if \(x>2\)

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure \(\PageIndex{21}\).

Figure \(\PageIndex{21}\): Graph of the entire function.

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

Graph the following piecewise function.

\[f(x)= \begin{cases} x^3 & \text{if $x < -1$} \\ -2 &\text{if $-1<x<4$} \\ \sqrt{x} &\text{if $x>4$} \end{cases} \nonumber\]

Figure \(\PageIndex{22}\)

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

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.
• The domain of a function can also be determined by identifying the input values of a function written as an equation.
• Interval values represented on a number line can be described using inequality notation, set-builder notation, and interval notation.
• For many functions, the domain and range can be determined from a graph.
• An understanding of toolkit functions can be used to find the domain and range of related functions.
• A piecewise function is described by more than one formula.
• A piecewise function can be graphed using each algebraic formula on its assigned subdomain.

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/Lea...s=MCRFPAK2&f=A .

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

a function in which more than one formula is used to define the output

a method of describing a set by a rule that all of its members obey; it takes the form \({x| statement about x}\)

Contributors

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a  Creative Commons Attribution License 4.0  license. Download for free at  https://openstax.org/details/books/precalculus .