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# Domain and Range of a Function

## Discrete and continuous functions and dependent and independent values

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Domain and Range of a Function

Credit: Larame Spence, CK-12 Foundation
Source: CK12.org

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*}.

#### 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

1. Define domain.
2. True or false? Range is the set of all possible inputs for the independent variable.
3. 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.

1. Dustin charges $10 per hour for mowing lawns. 2. Maria charges$25 per hour for math tutoring, with a minimum charge of 15. 3. \begin{align*}f(x) = 15x - 12\end{align*} 4. \begin{align*}f(x) = 2x^2 + 5\end{align*} 5. \begin{align*}f(x)=\frac{1}{x}\end{align*} 6. Make up a situation in which the domain is all real numbers but the range is all whole numbers. 7. 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? 8. 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? 9. Angie makes6.50 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.
10. 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.
11. 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.

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

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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.