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All courses Math II · F-BF.4.a 17 of 73
Find inverse functions by solving f(x)=c for simple invertible functions and writing inverse expressions.

Find the inverse of a linear function by swapping x and y and solving

Problem
Find the inverse of the linear function f(x)=2x+5.
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Hint

Write \(y=2x+5\), 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

\[y~=~2x+5\]

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

02

Swap x and y

\[x~=~2y+5\]

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

03

Solve for y

\[\begin{aligned} x-5~=~2y \\ y~=~(x-5)/2 \end{aligned}\]

Subtract 5 from both sides, then divide by 2.

04

Write the inverse

\[f^-1(x)~=~(x-5)/2\]

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

+

Another way

  1. You can think in reverse: \(2x+5\) means multiply by 2 and add 5, so the inverse subtracts 5 and then divides by 2.

!

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.