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

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Graph of an inverse function

The graph of $y = f^{- 1} (x)$ is the reflection of the graph of $y = f (x)$
- over the line $y = x$
This is because
- reflecting a point $(a, b)$ over the line $y = x$
- gives the point $(b, a)$
  - which is exactly the swap of inputs and outputs that the inverse performs
If a point $(a, b)$ lies on the graph of $f$ ,
- then the point $(b, a)$ lies on the graph of $f^{- 1}$
Key features of the graph also swap:
- $x$ -intercepts of $f$ become $y$ -intercepts of $f^{- 1}$
  - and vice versa
- Vertical asymptotes of $f$ become horizontal asymptotes of $f^{- 1}$
  - and vice versa
- If $f$ is increasing, then $f^{- 1}$ is also increasing

Graph showing three curves: red curve y=e^x, green curve y=ln x, and blue dashed line y=x, all plotted on Cartesian axes from -4 to 4. — **Graphs of the inverse functions e^x and lnx**

Examiner Tips and Tricks

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

to find $f^{- 1} (k)$ , 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.

Worked Example

Line graph with points at (0,0), (2,2), (3,6), and (4,8) on a 9x9 grid, marked axes labelled x and y, showing an increasing trend.

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) $\frac{1}{4}$

(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 $(3, 6)$ goes to $(6, 3)$
- point $(4, 8)$ goes to $(8, 4)$

The graph from the question, with the dashed line y=x added, along with line segments from (2,2) to (6,3), and from (6,3) to (8,4).

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

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

Graph of an inverse function

Examiner Tips and Tricks

Worked Example

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Author: Roger B

Reviewer: Mark Curtis

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

Graph of an inverse function

How is the graph of an inverse function related to the graph of the original?

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Author: Roger B

Reviewer: Mark Curtis