
Here we “undo” functions.

If a function maps every “input” to exactly one “output,” an inverse of that function maps every “output” to exactly one “input.” We need a more formal definition to actually say anything with rigor. These two simple equations are somewhat more subtle than they initially appear.
Let $f$ be a function. If the point $(1,9)$ is on the graph of $f$, what point must be the the graph of $f^{-1}$?
Which of the following is notation for the inverse of the function $\sin (\theta )$ on the interval $[-\pi /2,\pi /2]$?
$\sin ^{-1}(\theta )$ $\sin (\theta )^{-1}$
Consider the graph of $y=f(x)$ below Is $f(x)$ invertible at $x=1$?
yes no

So far, we’ve only dealt with abstract examples. Let’s see if we can ground this in a real-life context.

We have examined several functions in order to determine their inverse functions, but there is still more to this story. Not every function has an inverse, so we must learn how to check for this situation.

Let $f$ be a function, and imagine that the points $(2,3)$ and $(7,3)$ are both on its graph. Could $f$ have an inverse function?
yes no

Look again at the last question. If two different inputs for a function have the same output, there is no hope of that function having an inverse function. Why? This is because the inverse function must also be a function, and a function can only have one output for each input. More specifically, we have the next definition.

Which of the following are functions that are also one-to-one?
Mapping words to their meaning in a dictionary. Given a runner racing forward on a straight path, mapping time to the position of the runner. Mapping people to their birth date. Mapping mothers to their children.
Which of the following functions are one to one? Select all that apply.
$f(x) = x$ $f(x) = x^2$ $f(x) = x^3 - 4x$ $f(x) = x^3+4$

You may recall that a plot gives $y$ as a function of $x$ if every vertical line crosses the plot at most once, and we called this the vertical line test. Similarly, a function is one-to-one if every horizontal line crosses the plot at most once, and we call this the horizontal line test.

Below, we give a graph of $f(x)=-5x^2+30x+60$. While this graph passes the vertical line test, and hence represents $y$ as a function of $x$, it does not pass the horizontal line test, so the function is not one-to-one. Although the line $y=20$ intersects the curve $y=f(x)$ exactly once, the function $f$ is not one-to-one on its domain, since the line $y=80$ intersects the curve twice.

As we have discussed, we can only find an inverse of a function when it is one-to-one. If a function is not one-to-one, but we still want an inverse, we must restrict the domain. Let’s see what this means in our next examples.

Consider the graph of the function $f$ below: On which of the following intervals is $f$ one-to-one?
$[A,B]$ $[A,C]$ $[B,D]$ $[C,E]$ $[C,D]$

This idea of restricting the domain is critical for understanding functions like $f(x) = \sqrt {x}$.

Consider the graph of $y=f(x)$ below Is the function $f$ one-to-one on the interval $[1,2]$?
yes no
Let $f^{-1}$ be the inverse function of $f$ restricted to the domain $[1,2]$.
(a)
Find the domain of $f^{-1}$.
(b)
Find the value of $f\left ( f^{-1}(0.5)\right )$.
(c)
Choose the correct answer regarding $f^{-1}(0.5)$.
$f^{-1}(0.5)>1.5$ $f^{-1}(0.5)\le 1.5$ $f^{-1}(0.5)$ is not defined