Graph of an Inverse Function (College Board AP® Precalculus): Study Guide

When reading inverse values from a graph of $f$:

• to find $f^{-1}\left(k\right)$, locate $k$ on the $y$-axis (not the $x$-axis)

• then read across to the curve and down to the $x$-axis

A common mistake is to look up $k$ on the $x$-axis instead.

The graph of the piecewise-linear function $f$ is shown in the figure. Let $g$ be the inverse function of $f$. What is the maximum value of $g$?

(A)  $\frac{1}{8}$

(B)  $\frac{1}{4}$

(C)  $4$

(D)  $8$

Answer:

The graph of an function's inverse is a reflection of the graph of the function over the line $y=x$

• Sketch the reflection of the graph of $f$

  • point $\left(3,6\right)$ goes to $\left(6,3\right)$

  • point $\left(4,8\right)$ goes to $\left(8,4\right)$

You can see from the sketch that the maximum value of $g$ occurs at the point $\left(8,4\right)$

(C)  $4$