Finding an Inverse
Finding an Inverse
Polynomials that are strictly increasing or strictly decreasing have inverse functions. For example, and +x+1 are strictly increasing. However, neither nor x are onetoone, and so they do not have an inverse defined for all real .
3
x
5
x
2
x
3
x
x
A polynomial is onetoone 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 onetoone on those intervals and have inverses. For example, the function defined for with values has the inverse function . The function defined for with values has the inverse function .
x≥0
2
x
x
x≤0
2
x

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.