Inverses

To have an inverse that is a function it must pass the horizontal line test.

To find the inverse you switch the x and y and solve for y.

To prove something is an inverse do (f ᵒ f^-1)(x) and (f ^-1 ᵒ f)(y). Both should simplify to x.

F^-1(x) means inverse.

Ex 1. f(x) = x – ½ and g(x) = 2x + 1 Show that f(x) and g(x) are inverses

(f ᵒ g)(x) = f(g(x)) = 2x + 1 – 1/2 = x

(g ᵒ f)(x) = g(f(x)) = 2(x – 1/2) + 1 = x

Ex 2. Find the inverse of f(x) = 2x – 3

x = 2y – 3 y = x + 3/2

Ex 3. Find the inverse of y = sqrtx – 4

x^2 = sqrty + 4^2 x^2 = y – 4 y = x^2 – 4