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Section 7.2 Inverse of a Function

In this activity, we will introduce the inverse of a function.

A function is a rule which takes an input and returns an output. The inverse of a function is a rule which does the opposite of the original function — if the original function took the input \(a\) and returned the output \(b\text{,}\) then the inverse function would take the input \(b\) and return the output \(a\text{.}\)

Since a function is a process, the inverse function is just the reverse of that process.

In the following exercise, we introduce the notation for the inverse of a function.

Note 7.2.2. About the notation.

The expression \(f^{-1}(x)\) does not mean that something is being raised to the \(-1\) power. Rather, this is just special notation for the inverse of the function \(f\text{.}\)

Now, since a function \(f\) and its inverse \(f^{-1}\) do opposite things, we have a new way to describe certain function facts.

For example, the two statements below have the same meaning:

\begin{align*} f(3) \amp= 5\\ f^{-1}(5) \amp= 3 \end{align*}

If \(f\) takes the input \(3\) and outputs the number \(5\text{,}\) then \(f^{-1}\) will take the input \(5\) and output the number \(3\text{.}\)

Use this idea in the following exercise.

If \(f\) is a function, then \(f^{-1}\) should also be a function, and we know that these functions perfectly undo each other.

If we use the function \(f\) to get an output, and then use that output as the input for \(f^{-1}\text{,}\) we will get back the very input we originally used in the function \(f\text{.}\)

The diagram below shows this: It begins with an input value \(a\text{,}\) then uses the function \(f\) to get an output value \(f(a)\text{,}\) then uses the value \(f(a)\) as the input for the function \(f^{-1}\text{,}\) and finally gets back the original input \(a\text{.}\)

\begin{equation*} a\overset{f}{\longrightarrow}f(a)\overset{f^{-1}}{\longrightarrow}a \end{equation*}

Put another way, if we compose the functions \(f^{-1}\) and \(f\text{,}\) it will be as though nothing happened to the input.

In function notation, we would write:

\begin{equation*} f^{-1}(f(x)) = x \end{equation*}

Make sure you understand this concept in the next exercise.

By now, we know that the function \(f^{-1}\) will undo the action of \(f\text{.}\)

But there is another, equally important fact about composing a function and its inverse. See the next exercise.

Now let's use our understanding of inverse functions to interpret statements which use function notation.

Finding the formula for \(f^{-1}\).

If a function is relatively simple, it isn't hard to find its inverse.

For example, if \(f(x) = x + a\) then

\begin{equation*} f^{-1}(x) = x - a \end{equation*}

Or if \(g(x) = x\cdot b\) then

\begin{equation*} g^{-1}(x) = \frac{x}{b} \end{equation*}

These inverses are easy to find because we know that addition and subtraction are inverse operations, and that multiplication and division are inverse operations.

But how do we determine a formula for an inverse if the original function is more complicated?

Earlier in this activity, you discovered that:

\begin{equation*} f(f^{-1}(x)) = x \end{equation*}

In the next exercise, we will use this fact to find the formula for an inverse function.

Use the technique from the last problem to find the formula for an inverse function.

The graph of \(f^{-1}\).

Since \(f^{-1}\) is the reverse process of \(f\text{,}\) then the inputs and outputs have traded places.

In the next exercise, you will explore the the relationship between the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\text{.}\)

You may not have noticed it, but there was a nice geometric relationship between the graphs of \(f(x)\) and \(f^{-1}(x)\text{.}\) See the next problem to explore this relationship.

Not every function has an inverse.

Remember an important characteristic of any function:

Each input goes to only one output.

For example, if \(f\) is a function, then it would be impossible for both \(f(4) = 7\) and \(f(4) = 10\text{.}\) The input \(4\) cannot correspond to two different output values.

Thinking about the graph, this means that the graph of \(y = f(x)\) must pass the vertical line test, which says that a vertical line may only intersect the graph of a function at most once. If a vertical line intersected the graph of \(f(x)\) twice, that would mean there was some input value for which \(f\) gave two different output values — impossible for a function!

Now, an inverse function must also actually be a function. It, too, must pass the vertical line test. Each input value for the inverse must correspond to only one output value.

But what would cause this condition to fail for the inverse? Could we tell from the original function \(f\) whether the inverse was a function or not?

If a function \(f\) has an inverse which, itself, is not a function, we typically say that the function \(f\) is non-invertible. If the inverse for \(f\) is a function, then we say the function \(f\) is invertible.

Based on what you saw in the last problem, how could you tell from a function whether it is invertible or non-invertible?

This has implications for the graph of a function \(f\text{.}\) Given a graph of \(y = f(x)\text{,}\) we should be able to quickly decide if it is invertible or non-invertible.

In the last exercise, you saw that a function is invertible if no horizontal lines cross its graph more than once.

This is commonly referred to as the horizontal line test:

A function \(f\) is invertible if and only if no horizontal line intersects \(y = f(x)\) more than once.

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