3.4 inverse functions & relations. inverse relations two relations are inverses if and only if...

3.4 Inverse Functions & 3.4 Inverse Functions & RelationsRelations

Inverse RelationsInverse Relations

Two relations are inverses if and only if one Two relations are inverses if and only if one relation contains the element (b, a) whenever relation contains the element (b, a) whenever the other relation contains the element (a, b).the other relation contains the element (a, b).

If f(x) denotes a function then fIf f(x) denotes a function then f -1-1(x) denotes the (x) denotes the inverse. The inverse is not necessarily a inverse. The inverse is not necessarily a function. function.

Inverses are symmetric to each other with Inverses are symmetric to each other with respect to the line y = x. respect to the line y = x.

Ex 1Ex 1Graph f(x) = xGraph f(x) = x22 and it’s inverse. and it’s inverse.

Horizontal line test – used to determine if Horizontal line test – used to determine if the inverse of a relation will be a function.the inverse of a relation will be a function.

If every horizontal line intersects the graph If every horizontal line intersects the graph of the relation in at most one point, then of the relation in at most one point, then the inverse of the relation is a function.the inverse of the relation is a function.

Finding the inverse algebraically – Finding the inverse algebraically – 1. let y = f(x)1. let y = f(x)2. Interchange x and y2. Interchange x and y3. Solve the resulting equation for y.3. Solve the resulting equation for y.

Ex 2Ex 2f(x) = xf(x) = x22 - 4 - 4

Is the inverse a function?Is the inverse a function?

Find the inverse.Find the inverse.

Graph.Graph.

Ex 3Ex 3Graph Graph 1 2y x

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

Ex 4Ex 4

Given f(x) = 3xGiven f(x) = 3x22 + 7, find f + 7, find f-1-1(x) and verify that (x) and verify that f and ff and f-1-1 are inverse functions. are inverse functions.