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All courses Math III · F-BF.4.a 20 of 55
Find inverse functions for simple invertible functions, including rational examples.

Find an inverse function by swapping x and y and solving a linear equation

Problem
Find f⁻1(x) for f(x)=3x-5.
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Hint

Write \(y=3x-5\), then swap \(x\) and \(y\) before solving for \(y\).

To find an inverse, interchange the input and output variables and then solve for the new output.

Solution walkthrough

01

Write the function with y

\[y~=~3x-5\]

Start by rewriting the function using \(y\) instead of \(f(x)\).

02

Swap x and y

\[x~=~3y-5\]

Interchanging \(x\) and \(y\) reverses the roles of input and output.

03

Solve for y

\[\begin{aligned} x+5~=~3y \\ y~=~(x+5)/3 \end{aligned}\]

Add 5 to both sides, then divide by 3.

04

Write the inverse

\[f^-1(x)~=~(x+5)/3\]

Replace \(y\) with \(f^-1(x)\) to name the inverse function.

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Another way

  1. You can think in reverse: \(3x-5\) means multiply by 3 and subtract 5, so the inverse adds 5 and then divides by 3.

!

Common mistake

A common mistake is to change the signs without swapping the variables first. The inverse must undo the original operations in reverse order.