3.4 Inverse Functions & Relations

Inverse Relations Two relations are inverses if and only if one relation contains the element (b, a) whenever the other relation contains the element (a, b). If f(x) denotes a function then f-1(x) denotes the inverse. The inverse is not necessarily a function. Inverses are symmetric to each other with respect to the line y = x.

Ex 1 Graph f(x) = x2 and it’s inverse.

Horizontal line test – used to determine if the inverse of a relation will be a function. If every horizontal line intersects the graph of the relation in at most one point, then the inverse of the relation is a function.

Finding the inverse algebraically – 1. let y = f(x) 2 Finding the inverse algebraically – 1. let y = f(x) 2. Interchange x and y 3. Solve the resulting equation for y.

Ex 2 f(x) = x2 - 4 Is the inverse a function? Find the inverse. Graph.

Ex 3 Graph

Inverse functions – two functions, f and f-1, are inversed if and only if f(f-1(x)) = f-1(f(x)) = x

Ex 4 Given f(x) = 3x2 + 7, find f-1(x) and verify that f and f-1 are inverse functions.