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All courses Algebra I · F-BF.4.a 63 of 76
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

\[y=2x+5\]

Using y for the output makes it clear which variables will exchange roles when forming the inverse relation.

02

Swap input and output

\[x=2y+5\]

An inverse reverses every ordered pair, so interchange x and y in the original equation.

03

Solve for the new output

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

Subtract 5 from both sides, then divide the entire difference x-5 by 2. This isolates the new output y.

04

Name and verify the inverse

\[\begin{aligned} f^-1(x)=(x-5)/2 \\ f(f^-1(x))=2((x-5)/2)+5=x \\ f^-1(f(x))=((2x+5)-5)/2=x \end{aligned}\]

Both compositions simplify to x, so the derived rule reverses f in both directions. The correct inverse is (x-5)/2.

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Another way

  1. Reverse the operations in reverse order: subtract 5 from an output, then divide by 2.

!

Common mistake

Do not write x/2-5. Dividing after x-5 means both terms are divided by 2: (x-5)/2=x/2-5/2.

Solution walkthrough video