EZ Math

Precalculus · Functions and domains

How to Find the Inverse of a Linear Function

Problem

f(x)=2x+5f(x) = 2x + 5

Answer

f−1(x)=x−52f^{-1}(x) = \frac{x - 5}{2}

An inverse function undoes the original. Swap the input and output, then solve for the new output.

Step-by-step solution

  1. Write the function with y.

    y=2x+5y = 2x + 5

    Why: The inverse swaps the roles of the input and the output.

  2. Swap x and y.

    x=2y+5x = 2y + 5

    Why: Swapping turns the rule around: now y is the input.

  3. Solve for y.

    x−5=2y⇒y=x−52x - 5 = 2y \Rightarrow y = \frac{x - 5}{2}

    Why: Subtract 5, then divide by 2, the same steps in reverse.

  4. Rename y as the inverse function.

    f−1(x)=x−52f^{-1}(x) = \frac{x - 5}{2}

    Why: The result is the function that undoes f.

Common mistakes

  • Changing only one side when swapping. Swap every x with y and every y with x.
  • Forgetting to divide by 2 after subtracting 5.
  • Writing the inverse as 1/(2x + 5). The -1 is notation, not an exponent.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. f(x)=3x+6f(x) = 3x + 6

    Show answer

    f−1(x)=x−63f^{-1}(x) = \frac{x - 6}{3}

  2. f(x)=x−4f(x) = x - 4

    Show answer

    f−1(x)=x+4f^{-1}(x) = x + 4

  3. f(x)=4x−8f(x) = 4x - 8

    Show answer

    f−1(x)=x+84f^{-1}(x) = \frac{x + 8}{4}

  4. f(x)=5x+10f(x) = 5x + 10

    Show answer

    f−1(x)=x−105f^{-1}(x) = \frac{x - 10}{5}

  5. f(x)=2x−1f(x) = 2x - 1

    Show answer

    f−1(x)=x+12f^{-1}(x) = \frac{x + 1}{2}