How to Find the Range of a Function

Date: 02/25/98 at 16:44:12
From: Lynae Hunt
Subject: How to find the range

I'm a precalculus student at Elkhart Lake High School in Elkhart Lake, 
Wisconsin. My teacher and I are stumped on how to figure out the range 
for problems. An example of a problem is g(x) = (x+1)/(x^2-1). The 
book gives the range as

   (negative infinity, -1/2)U(-1/2, 0)U(0, infinity).

My question to you is how they got that range? How did they get 
the -1/2?

Thanks for the help

Date: 02/26/98 at 09:38:17
From: Doctor Anthony
Subject: Re: How to find the range

The range represents the set of values that y can take. You must 
exclude values which are impossible or undefined.

For example, if x = -1, y = 0/0.

                        x+1          1
We could write  y = ----------- =  -----, provided x NOT equal to -1.
                     (x+1)(x-1)     x-1

             1
However,   -----  = -1/2, as x -> -1
           -1-1

So y is tending to the value -1/2 but does not equal -1/2.

When x -> 1 from below, y -> -infinity; and when x -> 1 from above, 
y -> +infinity.

Finally, y = 0 is an asymptote, since the curve approaches y = 0 from 
below when x -> -infinity, and from above when x -> +infinity.

So y can take all values from -infinity to +infinity, but we exclude 
y = -1/2, where the curve is not defined; and exclude y = 0, which is 
an asymptote.

The statement you gave above for the range is simply expressing these 
facts.

-Doctor Anthony, The Math Forum
Check out our web site http://mathforum.org/dr.math/