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# 1.11: Domain and Range of a Function

Difficulty Level: Basic Created by: CK-12
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Suppose you have a function that allows you to input the number of years you have until retirement and which outputs the amount of money you should have saved. How would you go about determining the domain of such a function? How would you decide on the range? After completing this Concept, you'll be able to create a table of values for a function like this and give its domain and range.

### Watch This

Multimedia Link For a look at the domain of a function, see the following video where the narrator solves a sample problem from the California Standards Test about finding the domain of an unusual function. Khan Academy CA Algebra I Functions (6:34)

<iframe width='480' height='300' src='http://www.educreations.com/lesson/embed/1185292/?ref=app' frameborder='0' allowfullscreen></iframe>

### Guidance

Using a Function to Generate a Table

A function really is an equation. Therefore, a table of values can be created by choosing values to represent the independent variable. The answers to each substitution represent f(x)\begin{align*}f(x)\end{align*}.

#### Example A

Use Joseph’s function to generate a table of values. Because the variable represents the number of rides Joseph will pay for, negative values do not make sense and are not included in the list of values of the independent variable.

Solution:

R\begin{align*}R\end{align*} J(r)=2r\begin{align*}J(r) = 2r\end{align*}
0 2(0)=0\begin{align*}2(0) = 0\end{align*}
1 2(1)=2\begin{align*}2(1) = 2\end{align*}
2 2(2)=4\begin{align*}2(2) = 4\end{align*}
3 2(3)=6\begin{align*}2(3) = 6\end{align*}
4 2(4)=8\begin{align*}2(4) = 8\end{align*}
5 2(5)=10\begin{align*}2(5) = 10\end{align*}
6 2(6)=12\begin{align*}2(6) = 12\end{align*}

As you can see, the list cannot include every possibility. A table allows for precise organization of data. It also provides an easy reference for looking up data and offers a set of coordinate points that can be plotted to create a graphical representation of the function. A table does have limitations; namely it cannot represent infinite amounts of data and it does not always show the possibility of fractional values for the independent variable.

Domain and Range of a Function

The set of all possible input values for the independent variable is called the domain. The domain can be expressed in words, as a set, or as an inequality. The values resulting from the substitution of the domain represent the range of a function.

The domain of the function representing Joseph’s situation will not include negative numbers because it does not make sense to ride negative rides. He also cannot ride a fraction of a ride, so decimals and fractional values do not make sense as input values. Therefore, the values of the independent variable r will be whole numbers beginning at zero.

Domain: All whole numbers

The values resulting from the substitution of whole numbers are whole numbers times two. Therefore, the range of the function representing Joseph’s situation is still whole numbers, just twice as large.

Range: All even whole numbers

#### Example B

A tennis ball is bounced from a height and bounces back to 75% of its previous height. Write the function for this scenario and determine its domain and range.

Solution: The function representing this situation is h(b)=0.75b\begin{align*}h(b)= 0.75b\end{align*}, where b\begin{align*}b\end{align*} represents the previous bounce height.

Domain: The previous bounce height can be any positive number, so b0\begin{align*}b \ge 0\end{align*}.

Range: The new height is 75% of the previous height, and therefore will also be any positive number (decimal or whole number), so the range is all positive real numbers.

#### Example C

Find the range of f(x)=2x3\begin{align*}f(x)=2x-3\end{align*} when the domain is 0,1,2,3\begin{align*}0, 1, 2, 3 \end{align*}.

Solution:

Since the range is the output, we plug in the values in the domain to see what values we will get out.

f(0)=2(0)3=3\begin{align*}f(0)=2(0)-3=-3\end{align*}

f(1)=2(1)3=1\begin{align*}f(1)=2(1)-3=-1\end{align*}

f(2)=2(2)3=1\begin{align*}f(2)=2(2)-3=1\end{align*}

f(3)=2(3)3=3\begin{align*}f(3)=2(3)-3=3\end{align*}

The range for the given domain is 3,1,1,3\begin{align*}-3, -1, 1, 3\end{align*}.

Notice that we used function notation to keep track of which input value gave us which output value. This will be useful later.

Eli makes 20 an hour tutoring math. a. Write a function expressing the amount of money she earns. b. What are the domain and range of this function? c. Suppose Eli will only work for either 1, 1.5 or 2 hours. Express this domain and the corresponding range in a table. Solutions: a. Let M(h)\begin{align*}M(h)\end{align*} represent money earned for h\begin{align*}h\end{align*} hours. Then the function is M(h)=20h\begin{align*}M(h)=20h\end{align*}. b. Since hours worked can only be zero or positive, h0\begin{align*} h\ge 0\end{align*} is the domain. If Eli works for zero hours, she will earn zero dollars. She could also earn any positive amount of money, so the range is also all non-negative real numbers. That is, M0\begin{align*} M\ge 0\end{align*}. c. First we plug the domain into our function: M(1)=20(1)=20\begin{align*}M(1)=20(1)=20\end{align*} M(1.5)=20(1.5)=30\begin{align*}M(1.5)=20(1.5)=30\end{align*} M(2)=20(2)=40.\begin{align*}M(2)=20(2)=40.\end{align*} Putting this into a table, we get: h\begin{align*}h\end{align*} M(h)\begin{align*}M(h)\end{align*} 1 20 1.5 30 2 40 ### Practice Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Domain and Range of a Function (12:52) 1. Define domain. 2. True or false? Range is the set of all possible inputs for the independent variable. 3. Generate a table from 5x5\begin{align*}-5 \le x \le 5\end{align*} for f(x)=(x)22\begin{align*}f(x)= -(x)^2- 2\end{align*}. In 4-8, identify the domain and range of the function. 1. Dustin charges10 per hour for mowing lawns.
2. Maria charges $25 per hour for math tutoring, with a minimum charge of$15.
3. f(x)=15x12\begin{align*}f(x) = 15x - 12\end{align*}
4. f(x)=2x2+5\begin{align*}f(x) = 2x^2 + 5\end{align*}
5. f(x)=1x\begin{align*}f(x)=\frac{1}{x}\end{align*}
1. Make up a situation in which the domain is all real numbers but the range is all whole numbers.
2. What is the range of the function y=x25\begin{align*}y = x^2 - 5\end{align*} when the domain is 2\begin{align*}-2\end{align*}, 1\begin{align*}-1\end{align*}, 0, 1, 2?
3. What is the range of the function y=2x34\begin{align*}y = 2x - \frac{3}{4}\end{align*} when the domain is 2.5\begin{align*}-2.5\end{align*}, 1.5, 5?
4. Angie makes \$6.50 per hour working as a cashier at the grocery store. Make a table of values that shows her earnings for the input values 5, 10, 15, 20, 25, 30.
5. The area of a triangle is given by: A=12bh\begin{align*}A = \frac{1}{2}bh\end{align*}. If the base of the triangle is 8 centimeters, make a table of values that shows the area of the triangle for heights 1, 2, 3, 4, 5, and 6 centimeters.
6. Make a table of values for the function f(x)=2x+3\begin{align*}f(x) = \sqrt{2x + 3}\end{align*} for the input values 1\begin{align*}-1\end{align*}, 0, 1, 2, 3, 4, 5.

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### Vocabulary Language: English Spanish

domain

The domain of a function is the set of $x$-values for which the function is defined.

Range

The range of a function is the set of $y$ values for which the function is defined.

Continuous

Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.

dependent variable

The dependent variable is the output variable in an equation or function, commonly represented by $y$ or $f(x)$.

Discrete

A relation is said to be discrete if there are a finite number of data points on its graph. Graphs of discrete relations appear as dots.

Formula

A formula is a type of equation that shows the relationship between different variables.

Function

A function is a relation where there is only one output for every input. In other words, for every value of $x$, there is only one value for $y$.

independent variable

The independent variable is the input variable in an equation or function, commonly represented by $x$.

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.

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