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

1 Answer
Wataru
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.

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