Inverse Functions

An inverse function reverses the operation of the original function. If f(x) takes an input x and produces an output y, then the inverse function f⁻¹(y) takes y back to x.

1. Replace f(x) with y. 2. Switch x and y. 3. Solve for y. 4. Replace y with f⁻¹(x).

(2, 1) (8, 3) (6, -3)

f⁻¹(x) = x - 5

f⁻¹(x) = -1/4x - 3

f⁻¹(x) = (x + 5)/3

A function is one-to-one if it never assigns the same value to two different domain elements. This means that each output is produced by exactly one input.

The horizontal line test is a method to determine if a function is one-to-one. If any horizontal line intersects the graph of the function more than once, the function is not one-to-one.

If (a, b) is a point on the function f, then (b, a) is a point on its inverse f⁻¹.

To verify, check if f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all x in the domain.