Finding the Range of a Function, Algebraically

If you recall first learning about range many years ago, you may have thought about it as subtracting the lowest value from the highest value in a dataset. We will now be taking this a step further, looking at range of functions.

Finding the range of a function $y=f(x)$ is the same as finding all values that $y$ could be. To do this, we can think of it this way:

The range of $f(x)$ is all the $y$-values where there is a number $x$ with $y=f(x)$.

We can try and motivate how to find this with an example. Let's try to find the range of: $$f(x)= x^2+1$$ So we want to find $y$-values such that there is some $x$ where $y=x^2+1$. Suppose we want to check if $y=5$ is in the range of $f(x)$. Then, we want to check if there is an $x$-value such that $x^2+1=5$. We can solve this equation as follows: $$\begin{align} x^2+1&=5\\ x^2&=4\\ x &= \pm 2 \end{align}$$ So since either $x=2$ or $x=-2$ works, we know that $y=5$ is in the range of $f(x)$.

More generally, if we want to find the full range of $y=x^2+1$, we can solve for $x$ (taking the inverse of the function) to get $x=\sqrt{y-1}$. Then, the range of $f(x)$ is simply the domain of $\sqrt{y-1}$, because these are all the values where there is some $x$-value with $f(x) = y$. In this example, the domain of $\sqrt{y-1}$ is just all values where $y-1 \geq 0$, so $[1,\infty)$ is the range of $f(x) = x^2+1$.

Overall, the steps for algebraically finding the range of a function are:

1. Write down $y=f(x)$ and then solve the equation for $x$, giving something of the form $x = g(y)$.

2. Find the domain of $g(y)$, and this will be the range of $f(x)$. Note: if there are certain $x$-values that cannot be in the domain, their associated $y$-value cannot be in the range!

3. If you can't seem to solve for $x$, then try graphing the function to find the range.

Common functions and their ranges

Below, we can list a few common functions and the ranges they have. This will help you find the range of more complicated functions without having to do all the steps above.

$1$. $f(x) = |x|$. The range of $f(x)$ is $[0,\infty)$, which is all non-negative real numbers.

$2$. $f(x) = \ln(x)$. The range of $f(x)$ is $(-\infty,\infty)$, which is all real numbers. In fact, this range holds for any base for the log.

$3$. For $a > 0$ and $a \neq 1$, $f(x) = a^x$ has a range of $(0,\infty)$.

While graphing functions can help us find the range of a function, sometimes it's easier to find the range algebraically, using a different set of techniques and ideas. Recall that the range of a function is all possible values the function (or dataset) can take on.

When finding the range of a function algebraically, we usually have to solve at least one equation or inequality. This is because a number $y$ is in the range of the function $f$ only if there is a value $x$ such that $f(x) = y$ (and any such $x$ must be in the domain of $f$).

Example: Let's say we want to find the range of the function $f(x) = |x|.$ You might remember that the absolute value function is always nonnegative. For instance, if we plug a positive number into the function then $f$ returns that number; if we plug a negative number into the function then $f$ removes the negative sign. And finally, $f(0) = |0| = 0.$

But this alone doesn't tell us that the range of $|x|$ is $[0, \infty).$ We need to show that every nonnegative number is the absolute value of some number. To do this, let's pick a random nonnegative number $y,$ and then we want to show that there is a real number $x$ such that

$$y = |x|.$$

However, a lot of the hard work is already done from our understanding of the absolute value function. We know that if we plug a nonnegative number into the absolute value function, we get the same number back. Therefore we can set $x = y$ since $y$ is nonnegative, and have

$$|y| = y.$$

In other words, we can "access" any nonnegative number by using the absolute value function, so the range of $f(x) = |x|$ is $[0, \infty).$

That problem was relatively easy because we have a lot of intuition on how the absolute value function works already. As a reenforcement, answer this question.

Now we're going to move to a completely different problem that doesn't involve absolute value functions.

Example: Now we want to find the range of $f(x) = -x^4 + 1.$ This function is a polynomial, and so its domain is $(-\infty, \infty);$ the function is defined for any real value of $x$ we plug in.

Now we think about this function one piece at a time since there are multiple terms. $x^4$ is always nonnegative, so $-x^4$ is always non-positive. Notice further that $-x^4$ has a maximum of $0$ since we can plug $x = 0$ into $-x^4$ to get $0.$

If we add $1$ to get the definition of $f(x),$ this means the maximum of $f(x)$ is $1,$ where we plug in $x = 0$ to get this maximum. The maximum of a function, when it exists, is the "right-hand" part of the function's range, which means our range is $?, 1],$ where we have to figure out what the $?$ is, and whether to use a $($ or a $[.$

So here's what we do: let $y$ be a real number that is at most $1,$ and see which values of $y$ can be "reached" by at least one real number $x.$ This is the same thing as solving the equation

$$y = -x^4 + 1$$

for $x,$ for as many different values of $y$ as possible. This part we can do! The goal is to isolate $x,$ so subtract $1$ from both sides of the equation:

$$y - 1 = -x^4.$$

Now divide both sides by $-1:$

$$1 - y = x^4.$$

The left-hand side is nonnegative because $y \leq 1,$ and this remains true no matter how small $y$ gets. Therefore we can take the fourth root of both sides of the equation to get

$$\pm \sqrt[4]{1 - y} = x.$$

Since these values of $x$ are well-defined for any number $y \leq 1,$ $y$ could be a really, really small number. That is, the range is $(-\infty, 1].$ We always use a $($ when the "end" of a range is $-\infty$ because we say numbers can get as small as they want but never "reach" $-\infty.$

(Video) Finding the Range of a Function Algebraically

by Anusha Rahman

In this lesson, you will how to find the range of a function algebraically.

Summary

The range of a function is all possible $y$ values that a function can have. We can also think of range as the output values of a function.

There are four steps to algebraically find the range of a function:

1. Define the function in standard form, with "y = "

2. Express $x$ in terms of $y$

3. Find all possible values where $x=f(x)$ cannot be defined.

4. Write the range, being mindful of the values of $y$ that the function cannot take.

Finding Function Range: Algebraically

The find the range of a function algebraically, you first need to understand the values the function can output.

For example, if you have a function like $f(x)=x^2$, the output is always a positive number or zero because squaring any number gives a positive result or zero. To find the range, think about the smallest and largest values the function can product (the output). You could use inequalities or solve for $y$ to figure out which values are possible for the function's output. If you want to visualize the graph, look at the extrema.