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All courses Algebra II · F-BF.4.a 27 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 with y and swap variables

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

An inverse reverses input and output roles, so interchange x and y before solving.

02

Solve for the new output

\[x+5=3y~->~y=(x+5)/3\]

Add 5 and divide by 3 to isolate y.

03

Name and check the inverse

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

Composing gives f((x plus 5)/3)=x, confirming the inverse formula, choice A.

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Another way

  1. Reverse the original operations in reverse order: add 5, then divide by 3.

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Common mistake

After swapping x and y, solve for y completely; leaving x equals 3y minus 5 is not inverse function notation.

Solution walkthrough video