What Are Inverse Functions?

In a function, each input has one output. For any one-to-one function (only one value of $y$ for every $x$), the inverse function "undoes" the original function.

Another way to think about the inverse function is: if the function $f(x)$ gives the set of points $(x, f(x))$, the inverse function (denoted by $f^{-1}(x)$ gives the set of points $(f(x), x)$.

Thus, to invert the function basically means to switch the outputs and inputs.

Inverses of Compositions

In this explanation I'm going to talk about how to find the inverse of a composition of functions. Before going on you may want to read about simplifying composite functions and any of the materials here on inverses you'd like.

Suppose $f$ and $g$ are invertible functions. That is, there are functions $f^{-1}$ and $g^{-1}$ such that

$$f(f^{-1}(x)) = f^{-1}(f(x)) = x$$ $$g(g^{-1}(x)) = g^{-1}(g(x)) = x$$

whenever $x$ is in the domain of $f$ and $g.$ Now we're going to talk about inverses of compositions of functions.

A composition of two functions $f$ and $g$ is a single function represented by the expression $f(g(x)),$ meaning we have an input $x$ and then apply $g$ to it. Whatever output we get, we apply to $f.$ Note that this is DIFFERENT from $g(f(x)),$ which applies $f$ to the argument $x$ and then applies $g.$

Let's think about whether $f(g(x))$ has an inverse. Remember, our goal is to apply a function to get "back" to $x,$ and it's important that $f$ and $g$ have inverses. We can think of this as a game, where we try to "rescue" $x$ from inside $f(g(x)).$ Let's work our way from the outside in. If we apply $f^{-1},$ we get

$$f^{-1}(f(g(x)) = g(x).$$

Now, if we apply $g^{-1},$ we will get

$$g^{-1}(f^{-1}(f(g(x))) = g^{-1}(g(x)) = x.$$

This makes us think $g^{-1}(f^{-1}(x))$ is the inverse of $f(g(x)),$ and this will be true if we also check that $f(g(g^{-1}(f^{-1}(x)))).$ Let's do that one step at a time, using that $f^{-1}$ is the inverse of $f$ and $g^{-1}$ is the inverse of $g.$

$$f(g(g^{-1}(f^{-1}(x)))) = f(f^{-1}(x)) = x.$$

We used that $g(g^{-1}(y)) = y,$ where $y = f^{-1}(x).$ That let us simplify everything to the end.

In summary, if $f$ and $g$ have inverse functions $f^{-1}$ and $g^{-1},$ respectively, the inverse of $f(g(x))$ is $g^{-1}(f^{-1}(x)).$

Now let's do an example with actual functions. This "composition-inverse formula" can save some work because we don't have to find $f(g(x))$ directly first.

Example: If $f(x) = x - 3$ and $g(x) = x^3 + 1,$ find $(f(g(x))^{-1}.$

Solution: Our formula says that

$$(f(g(x)))^{-1} = g^{-1}(f^{-1}(x)),$$

so our first step is to find $f^{-1}(x).$ This is familiar territory since $f$ is a linear function.

$$f(x) = x - 3$$

Set $y = f(x),$ so

$$y = x - 3.$$

Now, flip the locations of $y$ and $x$ just like any other "find the inverse" problem.

$$x = y - 3.$$

Let's solve for $y$ again, which actually only takes one step: adding $3$ to both sides:

$$y = x + 3.$$

This means that $$g^{-1}(f^{-1}(x)) = g^{-1}(x + 3).$$ Now we need to find $g^{-1}(x)$ (that formula, when we find it, can have $x + 3$ substituted in to replace $x$).

Recall that

$$g(x) = x^3 + 1,$$

so just like when we found $f^{-1},$ set $y = g(x):$

$$y = x^3 + 1.$$

Now swap the $x$ and $y:$

$$x = y^3 + 1.$$

Let's solve for $y,$ which will require two steps. Subtract $1$ from both sides:

$$x - 1 = y^3.$$

Take the cube root of both sides:

$$\sqrt[3]{x - 1} = y$$

This means that $g^{-1}(x) = \sqrt[3]{x - 1}.$ Therefore,

$$g^{-1}(x + 3) = \sqrt[3]{(x + 3) - 1} = \sqrt[3]{x + 2}.$$

This function is $g^{-1}(f^{-1}(x)),$ which is $(f(g(x)))^{-1}.$

Inverse Functions

Inverse functions are functions where the $x$ and $y$ coordinates are flipped. That is essentially a reflection over the line $y=x$.

A function must be one-to-one in order to have an inverse.

Let's look at an example!

Example

Given the function $y=3\left(x+3\right)^{3}$, find the inverse.

To start, we are going to graph the original function, which we can see below in purple.

Next, since we know we are looking for the inverse of the function, we will add the graph of $y=x$, which we can see below in orange.

The orange line will be the line we will reflect our function over. You will learn more about the specifics of how to find the inverse graphically and algebraically. For now, I will tell you that the equation for the inverse function is $x=3\left(y+3\right)^{3}$. You can see the graph of the inverse function is green.

We can use the horizontal line test to check if the inverse graph is also a function.

Looking Further

A function must be both one-to-one and onto to have an inverse over its codomain. Being one-to-one means each output comes from only one input, so you can reverse it. Being onto means the function uses every possible output value. If a function doesn’t have both of these, you can’t fully reverse it.

(Video) Inverse Functions

by Professor Dave Explains

In this video, Professor Dave Explains walks you through what an inverse function is.

Summary

We know that when we put a value into a function (the input), we get another value (the output). If we plug this output value into the inverse function, we should get the input value back. Therefore, we can say that:

If a function $= f(x)$, the inverse function is NOT $\frac{1}{f(x)}$.

This definition is different than what we have previously seen for "inverse" in math. When we talked about inverses before, we talked about fraction inverses, which were essentially reciprocals. That is not the case for function inverses. We must use algebra to find the inverse of a function.

To find the inverse of a function, there are three main steps:

1. Change $f(x)$ into $y$.

2. Swap $y$ and $x$.

3. Solve for $y$.

We can express a function inverse as $f^{-1}(x)$. The inverse function must undo all of the operations from the original function.

Some key notes about inverse functions:

• The domain of $f(x) =$ the range of $f^{-1}(x)$, restricting the domain.

• The domain of $f^{-1}(x) =$ the range of $f(x)$, restricting the domain.

• If, after swapping $x$ and $y$, we can't solve, then the function does not have an inverse. If we are able to solve, then the function has an inverse.

• Plugging the output of a function into its inverse function will always yield the original input value.

• $f[f^{-1}(x)] = x$ AND $f^{-1}[f(x)]=x$ must both be true.

Try some of the examples at the end!