Functions & Inverses
What Is an Inverse Function?
An inverse function undoes the original function. If f sends an input to an output, f⁻¹ sends that output back to the original input.
Undoing a function
If f(2) = 7, then f⁻¹(7) = 2. The superscript −1 is part of the name of the inverse; it does not mean “one over f.”
How to read inverse statements
- Start from a known value f(a) = b.
- Swap to get f⁻¹(b) = a.
- Remember that f⁻¹ undoes f.
- Do not treat −1 as an ordinary exponent.
Worked example
If f(4) = 10, find f⁻¹(10)
- f sends 4 to 10.
- The inverse sends 10 back to 4.
f⁻¹(10) = 4.
Common mistakes
- Writing f⁻¹(4) = 10 instead of swapping.
- Reading f⁻¹ as 1/f.
- Assuming every formula automatically has an inverse without checking.
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