Functions & Inverses
Finding Inverse Functions
To find an inverse algebraically, write y = f(x), swap x and y, then solve for y. The resulting formula is f⁻¹(x).
Swap and solve
Swapping x and y reverses the input–output pairs. Solving for the new y produces the inverse formula. For f(x) = 2x + 3, the inverse is (x − 3)/2.
How to find f⁻¹(x)
- Write y = f(x).
- Swap x and y.
- Solve the equation for y.
- Rename y as f⁻¹(x).
- Check by composing if needed.
Worked example
Find the inverse of f(x) = 2x + 3
- Write y = 2x + 3.
- Swap: x = 2y + 3.
- Subtract 3: x − 3 = 2y.
- Divide by 2: y = (x − 3)/2.
f⁻¹(x) = (x − 3)/2.
Common mistakes
- Solving for x without swapping first.
- Forgetting to reverse addition or multiplication.
- Leaving the answer in terms of y instead of writing f⁻¹(x).
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