How do you find domain and range of a rational function?

1 Answer
Oct 30, 2014

The domain of a rational function is all real numbers that make the denominator nonzero, which is fairly easy to find; however, the range of a rational function is not as easy to find as the domain. You will have to know the graph of the function to find its range.


Example 1

#f(x)=x/{x^2-4}#

#x^2-4=(x+2)(x-2) ne 0 Rightarrow x ne pm2#,

So, the domain of #f# is

#(-infty,-2)cup(-2,2)cup(2,infty)#.

The graph of #f(x)# looks like:

enter image source here

Since the middle piece spans from #-infty# to #+infty#, the range is #(-infty,infty)#.


Example 2

#g(x)={x^2+x}/{x^2-2x-3}#

#x^2-2x-3=(x+1)(x-3) ne 0 Rightarrow x ne -1, 3#

So, the domain of #g# is:

#(-infty,-1)cup(-1,3)cup(3,infty)#.

The graph of #g(x)# looks like this:

enter image source here

Since #g# never takes the values #1/4# or #1#, the range of #g(x)# is
#(-infty,1/4)cup(1/4,1)cup(1,infty)#.


I hope that this was helpful.