Section 1.5Inverses of Functions

Box of Chocolates

Section D Section E Section F

Esther Tam Monica

The original function The inverse of the original function

Example1

Given

Write the equation for the inverse relation by interchanging the variables and transforming the equation so that y is in terms of x.

Plot the function and its inverse on the same screen, using equal scales for the two axes. Explain why the inverse relation is not a function.

Plot the line y = x on the same screen. How are the graph of the function and its inverse relation related to this line?

Parametric Equations

Using a third variable to relate x and y

This third variable is called the "parameter."

On your calculator, it's "t".

One value for the parameter yields values for both x and y

Plotting inverse images becomes particularly easy using parametric equations.

If set x equal to t, we can simply rewrite f(x) in terms of t and reproduce our original function as a pair of parametric equations.

Since the first step we take in determining an inverse relationship is switching our x and y variables in our function expression, we can obtain the same result by switching the x-t and y-t relationships in our parametric equations.

Example 2

Plot the graph of for x in the domain

and its inverse using parametric equations

(Domain and range of both?)

Example 3

Let

Find the equation of the inverse of f. Plot function f and its inverse on the same screen.

Is f an invertible function? Why or why not?

Quick test - horizontal line test

Note: invertible functions are called one-to-one functions. These are functions that are strictly increasing or strictly decreasing.

Show algebraically that the composition of with is

Definitions and Properties: Function Inverses

The inverse of a relation in two variables is formed by interchanging the two variables.

If the inverse of function f is also a function, then f is invertible.

If f is invertible and then you can write the inverse of f as

To plot the graph of the inverse of a function, either

Interchange the variables, solve for y, and plot the resulting equation(s),

Or

Use parametric mode

If f is invertible, then the compositions of areand provided x is in the domain of f and

is in the domain of

provided x is in the domain of and

is in the domain of

A one-to-one function is invertible. Strictly increasing or strictly decreasing functions are one-to-one functions.