audio read-through The Graph of an Inverse Function

One-to-one functions and inverse functions are explored in these earlier sections:

If a point $\,(a,b)\,$ is on the graph of a one-to-one function $\,f\,,$ then the point $\,(b,a)\,$ is on the graph of its inverse $\,f^{-1}\,.$ This is because the input/output roles for a function and its inverse are switched—an input to one is an output from the other.

As discussed below, the point $\,(b,a)\,$ is found by reflecting the point $\,(a,b)\,$ about the line $\,y = x\,,$ when graphed in a coordinate system where ‘$\,1\,$’ on the $x$-axis is the same as ‘$\,1\,$’ on the $y$-axis.

From this observation, we will see below that the graph of an inverse function is very easy to obtain if we have the graph of the original functionjust reflect the original graph about the line $\,y = x\,.$

Finding a Point $\,(b,a)\,$ from a Given Point $\,(a,b)\,$

finding a point (b,a) from the point (a,b)

The Graph of an Inverse Function

By definition, the graph of a function is the set of all its input/output pairs, as the inputs vary over the domain.

Thus: $$ \begin{align} &\cssId{s40}{\text{the graph of } f^{-1}}\cr &\qquad \cssId{s41}{= \{(x,f^{-1}(x))\ |\ x\in\text{dom}(f^{-1})\}}\ \end{align} $$

Since $\,x\,$ is a dummy variable, this set can also be written using the dummy variable $\,y\,$: $$ \begin{align} &\cssId{s44}{\text{the graph of } f^{-1}}\cr &\qquad \cssId{s45}{= \{(y,f^{-1}(y))\ |\ y\in\text{dom}(f^{-1})\}} \end{align} $$

Now, we have:

$$ \begin{align} &\cssId{s47}{\text{the graph of $\,f^{-1}\,$}}\cr\cr &\quad \cssId{s48}{= \{(y,f^{-1}(y))\ |\ y\in\text{dom}(f^{-1})\}}\cr &\qquad\ \ \cssId{s49}{\text{(definition of the graph of a function)}}\cr\cr &\quad \cssId{s50}{= \{(y,f^{-1}(y))\ |\ y\in\text{ran}(f)\}}\cr &\qquad\ \ \cssId{s51}{\text{(since $\text{ran}(f) = \text{dom}(f^{-1})$)}}\cr\cr &\quad \cssId{s52}{= \{(f(x),x)\ |\ x\in\text{dom}(f)\}} \end{align} $$

For the last step:

$y\in\text{ran}(f)\,$
if and only if
there exists $\,x\in\text{dom}(f)\,$ such that $\,f(x) = y$

And: $$ \begin{gather} f(x) = y\cr \text{if and only if}\cr x = f^{-1}(y) \end{gather} $$

Thus, the graph of $\,f^{-1}\,$ is precisely the set of points that make up the graph of $\,f\,,$ but with the coordinates switched!

We've already seen that ‘switching coordinates’ is accomplished by reflecting about the line $\,y = x\,$ in a coordinate system where the scales on the horizontal and vertical axes are identical. Thus, we have:

GRAPHING AN INVERSE FUNCTION

Let $\,f\,$ be a one-to-one function, so that it has an inverse $\,f^{-1}\,.$

The graph of $\,f^{-1}\,$ is found as follows:

  • Create a coordinate system where ‘$1$’ on the $x$-axis is the same as ‘$1$’ on the $y$-axis.
  • Graph $\,f\,$ in this coordinate system.
  • Reflect (‘mirror’) the graph of $\,f\,$ about the line $\,y = x\,.$
  • The reflected graph is the graph of $\,f^{-1}\,.$

Several different one-to-one functions and their reflections in the line $\,y = x\,$ are shown below:

a function and its inverse
a function and its inverse
a function and its inverse

Concept Practice