Try not to think of a rational function ( f(x)/g(x) ) as being something completely different. The process is the same except that now the threat of a vertical asymptote is more imminent. Your process should be as follows:
- Find intercepts (where x=0 or y=0)
- Check for vertical asymptotes (where the denominator, or in your case g(x) is zero)
- Check for horizontal asymptotes (what happens as x goes to infinity)
- Find critical points (where f'(x) = 0)
- Determine concavity / whether or not your critical points are mins, maxes, or neither
To address rational functions specifically, going from a function that has no denominator (your f) to one that does (your g) the main difference will be in step 2 where now you have a big vertical line in your graph that sort of sucks whatever the shape of f was into it.
But the point I wanted to make was to not focus on how dividing by a function affects another, but to just follow the same steps regardless and you'll be fine no matter what the function looks like.