Functions & Inverses
Checking Inverse Functions
Two functions are inverses when each undoes the other. The strongest algebraic check is composition: f(f⁻¹(x)) and f⁻¹(f(x)) should both simplify to x.
Composition returns the input
If f and g are inverses, then f(g(x)) = x and g(f(x)) = x. You can also spot-check with a number: applying f and then f⁻¹ should recover the original value.
How to check an inverse
- Substitute the candidate inverse into f.
- Simplify and look for x.
- If useful, also check f⁻¹(f(x)).
- Optionally test with a concrete number.
Worked example
Check f(x) = 2x + 3 and f⁻¹(x) = (x − 3)/2
- Compute f((x − 3)/2) = 2 · (x − 3)/2 + 3.
- Simplify to x − 3 + 3.
- The composition equals x.
The two formulas are inverses.
Common mistakes
- Checking only one composition direction when asked for a full proof.
- Stopping before simplifying the composition.
- Confusing adding 3 with multiplying by 3 when testing candidates.
Practice more
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