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?

### The Domain and Range of a Function

#### Using Tables to Represent Functions

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 \begin{align*}f(x)\end{align*}.

Recall from the previous concept that Joseph and his friends decided to go to a theme park where each ride costs $2.00. Joseph can represent the cost of \begin{align*}r\end{align*} rides at a theme park by the function \begin{align*}J(r)=2r\end{align*}.

Let's 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.

\begin{align*}R\end{align*} | \begin{align*}J(r) = 2r\end{align*} |
---|---|

0 | \begin{align*}2(0) = 0\end{align*} |

1 | \begin{align*}2(1) = 2\end{align*} |

2 | \begin{align*}2(2) = 4\end{align*} |

3 | \begin{align*}2(3) = 6\end{align*} |

4 | \begin{align*}2(4) = 8\end{align*} |

5 | \begin{align*}2(5) = 10\end{align*} |

6 | \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

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.

Let's once again take a look at Joseph's function about the cost of the rides at the theme park.

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

#### Let's find the domain and range of the following situations:

- A tennis ball is bounced from a height and bounces back to 75% of its previous height.

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

Domain: The previous bounce height can be any positive number, so \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.**

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

Domain: The domain is given to be 0, 1, 2, 3

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

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

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

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

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

Range: \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.

### Examples

#### Example 1

Earlier, you were asked to pretend that you had 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?

Since input is the number of years until retirement, the domain should not include negative numbers and the numbers should be whole numbers. Theoretically, the domain is all positive whole numbers but due to the lifespan of humans, the domain is more likely to be all positive whole numbers below 100. The range will all positive numbers since the amount you have saved should be positive and there can be decimals when talking about money. Depending on how much you make, the amount you have saved can be large.

#### For Examples 2-4, use the following information: Eli makes $20 an hour tutoring math.

#### Example 2

Write a function expressing the amount of money Eli earns.

Let \begin{align*}M(h)\end{align*} represent money earned for \begin{align*}h\end{align*} hours. Then the function is \begin{align*}M(h)=20h\end{align*}.

#### Example 3

What are the domain and range of the function from Example 2?

Since hours worked can only be zero or positive, \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, \begin{align*} M\ge 0\end{align*}.

#### Example 4

Suppose Eli will only work for either 1, 1.5 or 2 hours. Express this domain and the corresponding range in a table.

First we plug the domain into our function:

\begin{align*}M(1)=20(1)=20\end{align*}

\begin{align*}M(1.5)=20(1.5)=30\end{align*}

\begin{align*}M(2)=20(2)=40.\end{align*}

Putting this into a table, we get:

\begin{align*}h\end{align*} |
\begin{align*}M(h)\end{align*} |
---|---|

1 | 20 |

1.5 | 30 |

2 |
40 |

### Review

- Define domain.
- True or false? Range is the set of all possible inputs for the independent variable.
- Generate a table from \begin{align*}-5 \le x \le 5\end{align*} for \begin{align*}f(x)= -(x)^2- 2\end{align*}.

In 4-8, identify the domain and range of the function.

- Dustin charges $10 per hour for mowing lawns.
- Maria charges $25 per hour for math tutoring, with a minimum charge of $15.
- \begin{align*}f(x) = 15x - 12\end{align*}
- \begin{align*}f(x) = 2x^2 + 5\end{align*}
- \begin{align*}f(x)=\frac{1}{x}\end{align*}
- Make up a situation in which the domain is all real numbers but the range is all whole numbers.
- What is the range of the function \begin{align*}y = x^2 - 5\end{align*} when the domain is \begin{align*}-2\end{align*}, \begin{align*}-1\end{align*}, 0, 1, 2?
- What is the range of the function \begin{align*}y = 2x - \frac{3}{4}\end{align*} when the domain is \begin{align*}-2.5\end{align*}, 1.5, 5?
- 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.
- The area of a triangle is given by: \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.
- Make a table of values for the function \begin{align*}f(x) = \sqrt{2x + 3}\end{align*} for the input values \begin{align*}-1\end{align*}, 0, 1, 2, 3, 4, 5.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.11.