Graphing Reciprocal Functions

Date: 11/01/2001 at 23:13:32
From: Willa Ross
Subject: How do you graph reciprocal functions?

My teacher tried to explain to us what a reciprocal function is but I 
don't understand. I know what a reciprocal is but I don't know how to 
graph it.

Some questions we have are:

f(x)= __1__     f(x)= _1_ -2    f(x)= __1__ -4
       x-2             x               x+3  

I also have to know how to find the domain and range of these graphs. 

I read some of the things in the archives and it mentioned several 
times about inverse functions, but I am not sure if they are the same 
thing as a reciprocal function.

Thank you very much for your help.

Date: 11/02/2001 at 12:48:01
From: Doctor Peterson
Subject: Re: How do you graph reciprocal functions?

Hi, Willa.

You're right that an inverse function is not the same as a reciprocal 
function. Reciprocal functions are simply functions that include a 
reciprocal.

Most of the work of graphing will be just the usual: pick some points 
and plot them, then use your knowledge of how the function works to 
fill in between them. So what can we tell about how they work, in 
order to make this easier?

The first question is to find the domain. Remember, the domain (unless 
you are told otherwise) will consist of ALL numbers EXCEPT those that 
cause trouble. The only trouble you can have in evaluating any of 
these functions is to try to divide by zero; so the domain will 
exclude only the value of x for which the denominator is zero. In the 
first example, this means x=2. So you can start your graph by lightly 
drawing a vertical line at x=2, to show that there will be no value 
there; in a sense the graph will be infinite there (that is, it will 
be very large near there, and getting even bigger as you approach the 
line). It's called a vertical asymptote, if you have heard of that.

The next thing to look at is behavior when x gets very large; this 
will give the horizontal asymptote. When x is very large, 1/x gets 
very close to zero. So in your first example, f(x) will be close to 
zero for large x. Notice that when x is a large POSITIVE number, 
1/(x-2) will be a small positive number, so f(x) will be falling 
gradually toward zero from above; you can sketch that in at the right 
of your graph. When x is a large negative number, f(x) will be a small 
negative number, so you can sketch it rising toward zero from below as 
you go off to the left. Your other examples add something to zero when 
x is large, but the same ideas apply.

Now you've got the overall behavior. When x is just above 2, f(x) will 
be a large positive value, and will decrease continually as x rises, 
making a big swooping curve; when x is just below x, f(x) will be a 
huge negative number, and it will swoop up toward zero as you go off 
to the left. Choose a couple of points to plot to get a feel for just 
how fast it swoops, and you can sketch the graph.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/