What the inverse does
The inverse function f⁻¹ undoes what f does. If f maps an object a to an image b, then f⁻¹ maps b back to a. A function has an inverse only if it is one-to-one (each image comes from exactly one object). Key property: ff⁻¹(x) = f⁻¹f(x) = x, so applying a function and then its inverse returns the original number.
Finding f⁻¹(x)
Write y = f(x), make x the subject, then swap the letters so the answer is written in terms of x. To find f⁻¹(a) directly, simply solve the equation f(x) = a for x — no rearranging of the whole formula is needed. The domain and range also swap over: the domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. Composition confirms an inverse is correct, because fg(x) = x and gf(x) = x together prove that g = f⁻¹. This is often the quickest way to check your working: substitute your answer back and see whether the two functions cancel to give x.
Key formula
If f(x) = (x + b)/a then f⁻¹(x) = ax − b. General property: ff⁻¹(x) = x. To find f⁻¹(a), solve f(x) = a.
Worked example
Given f(x) = 2x + 5. (a) Find f⁻¹(x). (b) Hence find f⁻¹(11).
(a) Let y = 2x + 5 ⇒ x = (y − 5)/2, so f⁻¹(x) = (x − 5)/2.
(b) f⁻¹(11) = (11 − 5)/2 = 6/2 = 3. Check: f(3) = 2(3) + 5 = 11 ✓.
Remember
- Only one-to-one functions have inverses.
- The graph of f⁻¹ is the reflection of f in the line y = x.
- To find f⁻¹(a) quickly, solve f(x) = a rather than rearranging first.