What Is an Inverse Function?
Given f: A → B, an inverse function f⁻¹: B → A "reverses" f. If f(a) = b, then f⁻¹(b) = a.
f⁻¹ exists if and only if f is bijective.
Example: f(x) = 2x + 1
To find inverse: y = 2x + 1 → x = (y - 1)/2
f⁻¹(x) = (x - 1)/2
Verify: f(f⁻¹(x)) = 2((x-1)/2) + 1 = x ✓
f⁻¹(f(x)) = (2x+1-1)/2 = x ✓Finding the Inverse
Steps:
1. Write y = f(x)
2. Solve for x in terms of y
3. Swap x and y
4. Verify f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Example: f(x) = (3x - 2)/(x + 1)
y = (3x - 2)/(x + 1)
y(x + 1) = 3x - 2
xy + y = 3x - 2
xy - 3x = -y - 2
x(y - 3) = -(y + 2)
x = -(y + 2)/(y - 3) = (y + 2)/(3 - y)
f⁻¹(x) = (x + 2)/(3 - x)Properties of Inverse Functions
f(f⁻¹(x)) = x for all x in the codomain of f
f⁻¹(f(x)) = x for all x in the domain of f
(f⁻¹)⁻¹ = f
(f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹ (order reverses!)