WOLFRAM|DEMONSTRATIONS PROJECT

Finding an Inverse

Polynomials that are strictly increasing or strictly decreasing have inverse functions. For example,

3

x

and

5

x

+x+1

are strictly increasing. However, neither

2

x

nor

3

x

-x

are one-to-one, and so they do not have an inverse defined for all real

x

.

A polynomial is one-to-one on its intervals of increase and decrease. A restriction of the polynomial is a new function, with one of those intervals as its domain, whose values agree with the values of the polynomial on that interval. Those functions are one-to-one on those intervals and have inverses. For example, the function defined for

x≥0

with values

2

x

has the inverse function

x

. The function defined for

x≤0

with values

2

x

has the inverse function

-

x

.

The graphs of a function and its inverse are symmetric in the line

y=x

.

This Demonstration plots the graphs of each restricted function (solid curve) and its inverse (dashed curve) in matching colors.