6 7 Inverse Relations and Functions Learning goals

6. 7 Inverse Relations and Functions Learning goals • find the inverse of a relation or function

Inverse Relations & Functions If (a, b) is an ordered pair of a relation, then (b, a) is an ordered pair of its inverse. • The domain of the function is the range of the inverse AND the range of the function is the domain of the inverse. If both a relation and its inverse happen to be functions, they are inverse functions. • Not all inverses are functions. function f(x) domain range inverse f – 1 (x)

Ex 1 Find the inverse of the relation x -1 0 1 2 y -2 -1 -1 -2 Find the domain and range of the relation and its inverse. function f(x) domain range inverse f – 1 (x)

The inverse of a relation is the reflection in the line y = x of the graph.

Finding an Inverse Relation 1. 2. 3. 4. Rewrite f(x) as y Swap the x and y Solve for y Rewrite y as f – 1 (x)

Ex 2 Find the inverse: function f(x) D R inverse f – 1 (x) • Is the inverse a function? • Find the domain and range of the function and its inverse.

Linear Equations • The domain and range of all lines is all real numbers except if the lines are ______ or _______. • All lines are functions except _______ lines.

Ex 3 Find the inverse function f(x) D R inverse f – 1 (x)

Function Inverse function f(x) Is the inverse a function? inverse f – 1 (x) D R Find the domain and range of the function and its inverse.

• Which of the above are functions? • What makes the other NOT a function? • So, if ____, then NOT a function.

Ex 4 Find the inverse: Graph and find domain and range. Is the inverse a function? function f(x) D R inverse f – 1 (x)

Ex 5 Find the inverse: Graph and find domain and range. Is the inverse a function? function f(x) D R inverse f – 1 (x)

Ex 6 Find the inverse: Graph and find domain and range. Is the inverse a function? function f(x) D R inverse f – 1 (x) What can you say is true of all cubic functions?

Composition of inverse functions: If then are inverses,

Ex 7 Let . Find each value.