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

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:

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:

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.