Expert: Bobby Soltani - 10/18/2005

Question
There were some definitions and stuff I had for homework and I couldnt figure out this one.....

Explain in paragraph form how to determine from a graph if a function and its inverse are functions


Answer
Hi Jessica,

A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.

Existence of an Inverse
Some functions do not have inverse functions. For example, consider f(x) = x^2. There are two numbers that f takes to 4, f(2) = 4 and f(-2) = 4. If f had an inverse, then the fact that f(2) = 4 would imply that the inverse of f takes 4 back to 2. On the other hand, since f(-2) = 4, the inverse of f would have to take 4 to -2. Therefore, there is no function that is the inverse of f.

Look at the same problem in terms of graphs. If f had an inverse, then its graph would be the reflection of the graph of f about the line y = x.

Note that the reflected graph does not pass the vertical line test, so it is not the graph of a function.

This generalizes as follows: A function f has an inverse if and only if when its graph is reflected about the line y = x, the result is the graph of a function (passes the vertical line test). But this can be simplified. We can tell before we reflect the graph whether or not any vertical line will intersect more than once by looking at how horizontal lines intersect the original graph!

Horizontal Line Test
Let f be a function.

If any horizontal line intersects the graph of f more than once, then f does not have an inverse.

If no horizontal line intersects the graph of f more than once, then f does have an inverse.

The property of having an inverse is very important in mathematics, and it has a name.

Definition: A function f is one-to-one if and only if f has an inverse.

The following definition is equivalent, and it is the one most commonly given for one-to-one.

Alternate Definition: A function f is one-to-one if, for every a and b in its domain, f(a) = f(b) implies a = b


Let me know if you have any questions.

Bobby