What Is the Range of a Function?

What is Range?

Range goes hand in hand with domain. Whereas domain focuses on the $x$-values a function is defined on, range focuses on the $y$-values a function is defined on.

If a function is a vending machine you can think of the range as the different candy bars that come out. Finding the domain and range is basically looking at the inputs and outputs to the function.

But let's look at a graph to really understand this.

Finding Range Graphically

This is the same graph from the explanation on finding the domain but now we're focusing on the $y$-values. What are the $y$-values that this function takes? Remember, range can include both negative and positive numbers!

Our graph starts at $-1$ on the $y$-axis and goes to $2$ on the $y$-axis (the horizontal axis). Since there's an open circle on $-1$, we don't include it in the range.

Because the graph is only defined between $-1$ and $2$ on the $y$-axis, our range is only between $-1$ and $2$. More precisely, the range is: $$R:\,\,-1<y\le2$$

Notice we didn't say $$R:\,\,-1\,\,{\color{#8925ae}{\le}}\,\, y\le 2$$ because $-1$ is not included in the range!

Interval Notation

One way to write the range of a graph is by using interval notation.

We start from the bottom and write the intervals that $y$ is defined on. Use brackets, $\left[\quad\right]$, when the endpoints are included and parentheses, $\left(\quad\right)$, when the endpoints are excluded.

Using the graph above, the range would be: $$R:\,\,(-1,2]$$ Because $-1$ is not included we use a parentheses. $2$ is included so we use a bracket.

There are three more points that our example hasn't covered.

1. All real numbers. If our range is any real number value, we can say it spans from negative infinity to infinity, $(-\infty, \infty ).$ But more concisely, we use the symbol $\mathbb{R}$ to say the range is all real numbers.
2. Infinities. If our range spans continuously from a point to the bottom or top, we say it goes to negative infinity or positive infinity respectively. When using interval notation for infinity we always use parentheses, since infinity isn't a point. For example, we might write $(-\infty, -5)$ or $[24, \infty).$
3. Multiple intervals. If a graph has multiple segments you might need to use multiple intervals. We can use the union symbol, $\cup$, to join them together. So if a graph's range is defined from $-2$ to $1$ and then from $4$ to infinity, we can write: $R:\,\,[-2,1]\cup[4,\infty)$.

Finding Range Algebraically

Finding the range of a function algebraically requires a little more work than it did with domain, but the idea is the same.

Usually, you'll be given an equation that is solve for $y$. To find the range of function, you need to first solve the equation for $x$. Then, look for the $y$'s that would break the equation.

Currently, there are two cases to look out for:

1. Fractions. Fractions "break" if their denominator equals $0$.
2. Square roots. Square roots break if there is a negative value under the root. Also remember you can't take a root of a negative number.

Example 1

We want to find the range for $$y=\frac{3}{x}+2$$

The first step is to solve the equation for $x$ instead of $y$. $$\begin{align} y&=\frac3x+2\\ y-2&=\frac3x\\ x(y-2)&=3\\ x&=\frac{3}{y-2} \end{align}$$

Now we have a fraction whose denominator cannot equal $0$. We wan't to find the $y$-value that makes the denominator $0$ so we can exclude it from our range. $$\begin{align} \text{denominator}&\ne 0\\ y-2&\ne 0\\ y&\ne 2 \end{align}$$ When $y=2$, we get a $0$ in the denominator which breaks our function. Therefore, we exclude $2$ from our range.

Finally, we can write the range. We know it cannot be $2$. There are two intervals to define this: a portion from negative infinity to $2$ (not included) and a portion from $2$ (not included) to infinity. $$R:\,\,(-\infty,2)\cup(2,\infty)$$

Graphing this, we see we were correct!

Example 2

Let's look at an example where there's a square root. $$f(x)=x^2-1$$

First, we solve for $x$. $$\begin{align} f(x)&=x^2-1\\ y&=x^2-1\\ y+1&=x^2\\ \pm\sqrt{y+1}&=x \end{align}$$

With square roots, an equation "breaks" if there's a negative value under the root. So our value under the root has to be nonnegative. We can set up an inequality to see when that is. $$\begin{align} \text{under the root}&\ge0\\ y+1&\ge0\\ y&\ge-1 \end{align}$$ As long as $y$ is greater than or equal to $-1$, our equation has a range.

Finally, we can write the range. We know $y$ has to be greater than or equal to $-1$. $$R:\,\,[-1,\infty)$$

Graphing this, we see we were correct.

Solution

First, we solve for $x$. $$\begin{align} y&=\frac{3}{x^2}\\ x^2(y)&=3\\ x^2&=\frac3y\\ x&=\pm\sqrt{\frac3y} \end{align}$$ We have both a fraction and a square root.

Let's start with the fraction. Our equation breaks when the denominator is $0$. So, $$y\ne0$$

Next is the square root. Our function breaks when the value under the root is negative. $\frac3y$ can only be negative if $y$ is negative. So, $$y\ge0$$

Combining these gives us the range: $$y>0$$

Which we can write in interval notation as: $$R:\,\,(0,\infty)$$

You may be asked to determine the domain and range of functions with complex numbers, but we'll cover that later!

(Video) Introduction to Functions

by mathman1024

In this video, mathman1024 explains the basics of functions. He goes over what the range is and how you find it.

$\large{\text{Summary}}$

Relations can be a set of ordered pairs: $$\{(2,3),(4,1),(0,1),(2,-2),(-1,5)\}$$ Or they can be a continuous graph:

Range

When we talk about relations or functions, we want to talk about the domain and range. The range is the set of $y$-values used in the relation.

You can describe the range of a set by just listing the $y$ values of all the ordered pairs in the set.

In the example set above, $\{(2,3),(4,1),(0,1),(2,-2),(-1,5)\}$, the range is $R=\{-2,1,3,5\}$ (notice that there are multiple $1$'s in the relation but only one was written down for the domain).

What about the graph above? The range is all real numbers, since the $y$ values span from negative infinity to positive infinity. This can be written in a couple different ways: $R=(-\infty,\infty)$ or $R=\mathbb{R}$. The $\mathbb{R}$ symbol is used to represent the set of all real numbers.

The Range of a Function

The range of a function is all of its $y$-values.

Why would it be helpful to know the range of a function? You'll use the application of range when solving real-world application problems, such as when solving for the height of an object or solving for time.