Inverse Functions

One to one functions

Functions that have inverses Functions have inverses if f(x1) ≠ f(x2) when

x1 ≠ x2

Graphically you can use the horizontal line test to determine if a function is one to one

- no horizontal line will intersect the graph more than once if the function is one to one

Example: Determine if the following are one to one

f(x) = x3

f(x) = x2

Inverse Function f-1

f-1(y) = x f(x) = y

Domain of f-1 is the range of f

Range of f-1 is the domain of f

Example

If f(1) = 5, f(3) = 7, and f(8) = -10, find f-1(7), f-1(5), and f-1(-10)

Example

Find the inverse of f(x) = x3 + 2

Drawing the Inverse

The graph of f-1 is obtained by reflecting the graph of f about the line y = x

On calculator plot f, then use “DRAW” menu, #8 (DrawInv)

Example:

Draw inverse of f(x) = √(-1 – x)

Example

Show that the function f(x) = √(x3 + x2 + x + 1) is one to one for both f and f-1