Inverse Functions

An inverse function reverses what the original did. If f turns 3 into 10, then f⁻¹ turns 10 back into 3.

To find one:
1. Write y = f(x).
2. Swap x and y.
3. Solve for y.
4. Write the result as f⁻¹(x).

For f(x) = 2x + 5: swap to get x = 2y + 5, solve to get y = (x − 5)/2.

The graphs are mirror images across the line y = x.

Important: f⁻¹ does not mean 1/f. That superscript is notation for "inverse," not an exponent.

Only one-to-one functions have inverses. y = x² does not, unless you restrict it to x ≥ 0.

The inverse f⁻¹ satisfies f⁻¹(f(x)) = x and f(f⁻¹(x)) = x on the appropriate domains, and this composition test is the definitive verification.

A function has an inverse exactly when it is one-to-one (injective), which the horizontal line test detects graphically. Non-injective functions can be inverted on a restricted domain — the convention that √x is the inverse of x² depends on restricting to x ≥ 0, and the same device defines the inverse trigonometric functions.

Domain and range swap: dom(f⁻¹) = ran(f) and ran(f⁻¹) = dom(f). This is the fastest way to state the inverse's domain without re-deriving it, and it explains why the exponential and logarithm exchange their asymptotes.

The reflection across y = x is the geometric consequence of swapping coordinates, so any point (a, b) on f corresponds to (b, a) on f⁻¹.

The notation clash with reciprocals is genuinely unfortunate: f⁻¹(x) is the inverse function while [f(x)]⁻¹ is the reciprocal, and they are almost never equal.