Question

How do I find the Range of a RATIONAL Function?

Answer 1

for example \[\frac{ 2 }{ x ^{2} -2x-3}\]

Answer 2

For the most part, the only way to find the range of a function is by graphing it(you can sometimes use inverse functions, but it's usually much simpler to sketch it).

What does the function look like?

Factor out the denominator. The function has vertical asymptotes at x=-1 and x=3

\[\frac{ 2 }{ (x+1)(x-3) }\]

Let's evaluate points in the middle of those. Pick any you like, just determine whether or not the function is positive or negative between these values.



Recall that a function cannot change signs unless it "crosses" a vertical asymptote(horrible way to describe it, don't ever say that) or unless it has a 0. When does the function equal 0? Never!. So, we know that it is always under the x axis from -1 to 3, and the opposite on the other sides of the asymptote.

Using this information, let's graph the function.



That's a very rough sketch, but you can tell that the function takes on all values of y. So the range is all real numbers, or infinity to negative infinity.