Inverse function calculator

How to find an inverse function

1. Replace f(x) with y: y = 2x + 3

2. Swap x and y: x = 2y + 3

3. Solve for y: y = (x − 3)/2

4. Write as f⁻¹(x) = (x − 3)/2

Verification

To verify: f(f⁻¹(x)) should equal x, and f⁻¹(f(x)) should also equal x.

f(f⁻¹(x)) = 2((x−3)/2) + 3 = (x−3) + 3 = x ✓

When does the inverse exist?

A function has an inverse only if it is one-to-one (passes the horizontal line test — no horizontal line intersects the graph more than once). Functions like x² are not one-to-one on all reals, but are one-to-one when restricted to x ≥ 0.

Common inverse functions

f(x) = x + b → f⁻¹(x) = x − b

f(x) = ax → f⁻¹(x) = x/a

f(x) = eˣ → f⁻¹(x) = ln(x)

f(x) = sin(x) → f⁻¹(x) = arcsin(x) (restricted domain)