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I'm taking a test soon, where I would be asked to sketch graphs. I wonder if there is any kind of guide or general tutorial on the net how to carry out sketching. For instance I would much like to know how would you sketch $f(x)=\frac{ln (x)} {x}$ and $g(x) = \frac{e^x}{x}$ or in general if I have a function $f(x)$ the graph of which I know, how should I sketch $f(x)/x$ or an even more complex question:

If I am given the sketch of two functions $f(x), g(x)$ how may I sketch $\frac{f(x)}{g(x)}$?

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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:

  1. Find intercepts (where x=0 or y=0)
  2. Check for vertical asymptotes (where the denominator, or in your case g(x) is zero)
  3. Check for horizontal asymptotes (what happens as x goes to infinity)
  4. Find critical points (where f'(x) = 0)
  5. 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.

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