Introduction

We know that if and are two real numbers, then makes sense, as long as is not equal to zero. Let’s look at what happens when we make divisions by numbers very close to zero, but not equal to zero. Take for simplicity.

This pattern makes us want to say that equals to (whatever means, at this point), but this doesn’t work. To understand why, let’s consider divisions by numbers very close to zero, but this time negative.

The same reasoning as before would tempt us to say that equals . And this raises the question of whether or is the better choice. While on an instinctive psychological level we could think that is better than , there’s really no way to decide1 — and this turns out to be related to the concept of limit, which you’ll learn in Calculus.

Graph and End Behavior

To continue our discussion in a more precise way, let’s consider the function , defined for all real numbers except for zero, given by . This is a very famous function, particularly useful as the building block for rational functions, which we’ll discuss soon. Note that essentially what we have just done in the introduction was to consider the values as well as To get a good idea of the behavior a function has, our main strategy so far has been to just consider its graph. Naturally, plugging a handful of values won’t cut it. Let’s see what happens when we go to the other extreme and make divisions by very large numbers:

And from the negative side:

Here’s what the graph looks like.

Here’s what we can immediately see from the graph, confirming our intuition from the several divisions previously done:

We say that the line is a vertical asymptote for , while the line is a horizontal asymptote. We will discuss asymptotes of rational functions in general in the next unit.

Next, as far as symmetries go, we can see that the graph is symmetric about the origin. This indicates that is an odd function. You can also see this algebraically via

You may also noticed that the graph of is symmetric across the line . This indicates that is it’s own inverse! You can also see this algebraically via .

By the way, the graph of is called a hyperbola.