What if you were given a rule that relates two variables, like \begin{align*}f(x) = 5x^2 + 1\end{align*}? How could you find the domain and range of the function defined by that rule? After completing this Concept, you'll be able to identify the domain and range of functions like this one.

### Watch This

CK-12 Foundation: 0110S Introduction to Functions

For another 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

### Guidance

A **function** is a rule for relating two or more variables. For example, the price you pay for phone service may depend on the number of minutes you talk on the phone. We would say that the cost of phone service is a *function* of the number of minutes you talk. Consider the following situation.

*Josh goes to an amusement park where he pays $2 per ride.*

There is a relationship between the number of rides Josh goes on and the total amount he spends that day: To figure out the amount he spends, we multiply the number of rides by two. This rule is an example of a **function**. Functions usually—*but not always*—are rules based on mathematical operations. You can think of a function as a box or a machine that contains a mathematical operation.

Whatever number we feed into the function box is changed by the given operation, and a new number comes out the other side of the box. When we input different values for the number of rides Josh goes on, we get different values for the amount of money he spends.

The input is called the **independent variable** because its value can be any number. The output is called the **dependent variable** because its value depends on the input value.

Functions usually contain more than one mathematical operation. Here is a situation that is slightly more complicated than the example above.

*Jason goes to an amusement park where he pays $8 admission and $2 per ride.*

The following function represents the total amount Jason pays. The rule for this function is "multiply the number of rides by 2 and add 8."

When we input different values for the number of rides, we arrive at different outputs (costs).

These flow diagrams are useful in visualizing what a function is. However, they are cumbersome to use in practice. In algebra, we use the following short-hand notation instead:

\begin{align*}& \quad \ input\\ & \quad \ \ \ \downarrow\\ & \quad \underbrace{f(x)}= y \leftarrow output\\ & \ function\\ & \quad \ \ box\end{align*}

First, we define the variables:

\begin{align*}x =\end{align*} the number of rides Jason goes on

\begin{align*}y =\end{align*} the total amount of money Jason spends at the amusement park.

So, \begin{align*}x\end{align*} represents the input and \begin{align*}y\end{align*} represents the output. The notation \begin{align*}f()\end{align*} represents the function or the mathematical operations we use on the input to get the output. In the last example, the cost is 2 times the number of rides plus 8. This can be written as a function:

\begin{align*}f(x) = 2x + 8\end{align*}

In algebra, the notations \begin{align*}y\end{align*} and \begin{align*}f(x)\end{align*} are typically used interchangeably. Technically, though, \begin{align*}f(x)\end{align*} represents the function itself and \begin{align*}y\end{align*} represents the output of the function.

**Identify the Domain and Range of a Function**

In the last example, we saw that we can input the number of rides into the function to give us the total cost for going to the amusement park. The set of all values that we can use for the input is called the **domain** of the function, and the set of all values that the output could turn out to be is called the **range** of the function. In many situations the **domain** and **range** of a function are both simply the set of all real numbers, but this isn’t always the case. Let's look at our amusement park example.

#### Example A

Find the domain and range of the function that describes the situation:

*Jason goes to an amusement park where he pays $8 admission and $2 per ride.*

**Solution**

Here is the function that describes this situation:

\begin{align*}f(x) = 2x + 8 = y\end{align*}

In this function, \begin{align*}x\end{align*} is the number of rides and \begin{align*}y\end{align*} is the total cost. To find the domain of the function, we need to determine which numbers make sense to use as the input \begin{align*}(x)\end{align*}.

- The values have to be zero or positive, because Jason can't go on a negative number of rides.
- The values have to be integers because, for example, Jason could not go on 2.25 rides.
- Realistically, there must be a maximum number of rides that Jason can go on because the park closes, he runs out of money, etc. However, since we aren’t given any information about what that maximum might be, we must consider that all non-negative integers are possible values regardless of how big they are.

**Answer** For this function, the domain is the set of all non-negative integers.

To find the range of the function we must determine what the values of \begin{align*}y\end{align*} will be when we apply the function to the input values. The domain is the set of all non-negative integers: {0, 1, 2, 3, 4, 5, 6, ...}. Next we plug these values into the function for \begin{align*}x\end{align*}. If we plug in 0, we get 8; if we plug in 1, we get 10; if we plug in 2, we get 12, and so on, counting by 2s each time. Possible values of \begin{align*}y\end{align*} are therefore 8, 10, 12, 14, 16, 18, 20... or in other words all even integers greater than or equal to 8.

**Answer** The range of this function is the set of all even integers greater than or equal to 8.

#### Example B

*Find the domain and range of the following functions.*

a) A ball is dropped from a height and it bounces up to 75% of its original height.

b) \begin{align*}y = x^2\end{align*}

**Solution**

a) Let’s define the variables:

\begin{align*}x =\end{align*} original height

\begin{align*}y =\end{align*} bounce height

A function that describes the situation is \begin{align*}y = f(x) = 0.75x\end{align*}. \begin{align*}x\end{align*} can represent any real value greater than zero, since you can drop a ball from any height greater than zero. A little thought tells us that \begin{align*}y\end{align*} can also represent any real value greater than zero.

**Answer**

The domain is the set of all real numbers greater than zero. The range is also the set of all real numbers greater than zero.

b) Since there is no word problem attached to this equation, we can assume that we can use any real number as a value of \begin{align*}x\end{align*}. When we square a real number, we always get a non-negative answer, so \begin{align*}y\end{align*} can be any non-negative real number.

**Answer**

The domain of this function is all real numbers. The range of this function is all non-negative real numbers.

In the functions we’ve looked at so far, \begin{align*}x\end{align*} is called the **independent variable** because it can be any of the values from the domain, and \begin{align*}y\end{align*} is called the **dependent variable** because its value depends on \begin{align*}x\end{align*}. However, any letters or symbols can be used to represent the dependent and independent variables. Here are three different examples:

\begin{align*}y &= f(x) = 3x\\ R &= f(w) = 3w\\ v &= f(t)= 3t\end{align*}

These expressions all represent the same function: a function where the dependent variable is three times the independent variable. Only the symbols are different. In practice, we usually pick symbols for the dependent and independent variables based on what they represent in the real world—like \begin{align*}t\end{align*} for time, \begin{align*}d\end{align*} for distance, \begin{align*}v\end{align*} for velocity, and so on. But when the variables don’t represent anything in the real world—or even sometimes when they do—we traditionally use \begin{align*}y\end{align*} for the dependent variable and \begin{align*}x\end{align*} for the independent variable.

**Make a Table For a Function**

A table is a very useful way of arranging the data represented by a function. We can match the input and output values and arrange them as a table. For example, the values from Example 1 above can be arranged in a table as follows:

\begin{align*}& x \quad 0 \quad 1 \quad \ \ 2 \quad \ 3 \quad \ \ 4 \quad \ 5 \quad \ \ 6\\ & y \quad 8 \quad 10 \quad 12 \quad 14 \quad 16 \quad 18 \quad 20\end{align*}

A table lets us organize our data in a compact manner. It also provides an easy reference for looking up data, and it gives us a set of coordinate points that we can plot to create a graph of the function.

#### Example C

*Make a table of values for the function* \begin{align*}f(x) = \frac{1}{x}\end{align*}. Use the following numbers for input values: -1, -0.5, -0.2, -0.1, -0.01, 0.01, 0.1, 0.2, 0.5, 1.

**Solution**

Make a table of values by filling the first row with the input values and the next row with the output values calculated using the given function.

\begin{align*}& x \qquad \qquad -1 \quad -0.5 \quad -0.2 \ \ -0.1 \ \ -0.01 \quad 0.01 \quad 0.1 \quad \ 0.2 \quad \ 0.5 \quad 1\\ & f(x) = \frac{1}{x} \quad \frac{1}{-1} \quad \frac{1}{-0.5} \quad \frac{1}{-0.2} \quad \frac{1}{-0.1} \quad \frac{1}{-0.01} \quad \frac{1}{0.01} \quad \frac{1}{0.1} \quad \frac{1}{0.2} \quad \frac{1}{0.5} \quad \frac{1}{1}\\ & y \qquad \qquad -1 \quad -2 \qquad -5 \quad -10 \quad \ -100 \quad 100 \quad \ 10 \qquad 5 \qquad 2 \quad \ \ 1\end{align*}

When you’re given a function, you won’t usually be told what input values to use; you’ll need to decide for yourself what values to pick based on what kind of function you’re dealing with. We will discuss how to pick input values throughout these lessons.

Watch this video for help with the Examples above.

CK-12 Foundation: Introduction to Functions

### Vocabulary

A **function** is a rule for relating two or more variables, one of which is the input variable and the other is the output variable. The input is called the **independent variable** because its value can be any number. The output is called the **dependent variable** because its value depends on the input value. The set of all values that we can use for the input is called the **domain** of the function, and the set of all values that the output could turn out to be is called the **range** of the function.

### Guided Practice

*Identify the domain and then make a table of values for the function* \begin{align*}f(x) = \frac{1}{\sqrt{x}}\end{align*}. Use the following numbers for input values: 0.01, 0.16, 0.25, 1, 4.

**Solution**

Since you cannot compute the square root of negative numbers, these cannot be in the domain. Since we cannot have 0 in the denominator, 0 is also not in the domain. This means that the domain is all real numbers greater than zero.

Make a table of values by filling the first row with the input values and the next row with the output values calculated using the given function.

\begin{align*}& x \qquad \qquad 0.01 \quad 0.16 \quad \ 0.25 \quad \ 1 \quad 4\\ & f(x) = \frac{1}{x} \quad \frac{1}{\sqrt{0.01}} \quad \frac{1}{\sqrt{0.16}} \quad \frac{1}{\sqrt{0.25}} \quad \frac{1}{\sqrt{1}} \quad \frac{1}{\sqrt{4}}\\ & y \qquad \qquad 10 \quad \ 2.5 \qquad 2 \qquad 1 \quad \ \ 0.5\end{align*}

### Explore More

For 1-6, identify the domain and range of the following functions.

- Dustin charges $10 per hour for mowing lawns.
- Maria charges $25 per hour for tutoring math, 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*}
- \begin{align*}f(x) = \sqrt[3]{x}\end{align*}
- What is the range of the function \begin{align*}y = x^2 - 5\end{align*} when the domain is -2, -1, 0, 1, 2?
- What is the range of the function \begin{align*}y = 2x - \frac{3}{4}\end{align*} when the domain is -2.5, -1.5, 5?
- What is the domain of the function \begin{align*}y = 3x\end{align*} when the range is 9, 12, 15?
- What is the range of the function \begin{align*}y = 3x\end{align*} when the domain is 9, 12, 15?
- Angie makes $6.50 per hour working as a cashier at the grocery store. Make a table that shows how much she earns if she works 5, 10, 15, 20, 25, or 30 hours.
- The area of a triangle is given by the formula \begin{align*}A = \frac{1}{2}bh\end{align*}. If the base of the triangle measures 8 centimeters, make a table 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{2}x + 3\end{align*} for input values -1, 0, 1, 2, 3, 4, 5.