1.9 Inverse Functions f-1(x) Inverse functions have symmetry about the line y = x Inverse of ordered pairs. Ex. f(x) = {(1,4), (2,-4), (7,3)} The inverse is: f-1(x) = {(4,1), (-4,2), (3,7)} Note that the x and y’s are switched.

Definition of the Inverse of a Function f and g are inverse functions if f(g(x)) = x and g(f(x)) = x Ex. Show that the following functions are inverses of each other. f(x) = 2x3 -1 f(g(x)) = = x + 1 – 1 = x

y = x

One-to-one Function: A 1-1 function states that for every x there is exactly one y and for every y there is exactly one x. If a function is 1-1, then f(a) = f(b) implies that a = b Ex. f(x) = x3 + 1 Ex. f(x) = x2 - x a3 + 1 = b3 + 1 a2 – a = b2 - b a3 = b3 There’s no way to get a = b. a = b

A 1-1 function must pass both the vertical and horizontal line tests. Pg. 96 please To find an inverse of a function f, use the following procedure. 1. Interchange the x’s and y’s 2. Solve the equation for y 3. Replace f with f-1 in the new equation

Ex. Find the inverse of 2x = 5 – 3y 3y = 5 – 2x

The graph of f(x) and f-1(x) 2 1 1 2 y = x