**Question Corner and Discussion Area**

How do you graph an inverse function once you have solved it?It is actually easier to graph the inverse of a function than it is to solve for it.

First let's think about what it means to graph a function.
Suppose that we wanted to graph a function *f*(*x*).
To do this, we would substitute numerical values for *x* and
plot those ordered pairs (*x*,*y*) for which *y*=*f*(*x*).
For instance, if , we might
try plotting the points (0,0), (1,1), (2,4), (3,9), etc.

Now how would we plot the inverse of a function? If *g* is the inverse
function of *f*, then to graph *g* we'd plot the points (*x*,*y*)
where *y*=*g*(*x*). This condition is the same as the condition *x*=*f*(*y*),
so the graph of the inverse consists of points (*f*(*y*),*y*).

These are the same as the points on the graph of *f* but with the
order of the coordinates interchanged:
instead of plotting (*x*,*f*(*x*)) for various numerical values of
*x*, we plot (*f*(*y*),*y*) for various numerical values of *y*
(which is the same thing as plotting (*f*(*x*),*x*) for various numerical
values of *x*).
For example, some of the points on the graph of the inverse of
are (0,0), (1,1), (4,2), and (9,3).
Geometrically, we have just reflected the graph of the function
*f*(*x*) through the line *y*=*x* to get the graph of the inverse of *f*(*x*).

This method of graphing the "inverse" of a function always works, even
when the function doesn't have an inverse.
If the function doesn't have an inverse, it is because there are two
distinct values *a* and *b* which we can assign to *x* to get the
same value for *f*(*x*).
If we examine our function we will note that
*f*(2) = *f*(-2) = 4.
The corresponding points on the graph of our "inverse
function" are (4,2) and (4,-2).
Thus the graph which we constructed in this method is not really the
graph of a function, since the value of the inverse of *f*(*x*) is
not well defined at 4 (it could either be 2 or -2).

Even though this approach will not always give us the graph of a function,
it will whenever the inverse of *f*(*x*) exists.

This part of the site maintained by (No Current Maintainers)

Last updated: April 19, 1999

Original Web Site Creator / Mathematical Content Developer: Philip Spencer

Current Network Coordinator and Contact Person: Joel Chan - mathnet@math.toronto.edu

Go backward to Multiplying Matrices

Go up to Question Corner Index

Go forward to Does Every Function Have an Antiderivative?

Switch to text-only version (no graphics)

Access printed version in PostScript format (requires PostScript printer)

Go to University of Toronto Mathematics Network
Home Page