Question

How do you graph the rational function `f(x)=6/(x^2+x-2)`?

Answer 1

I would factorize the denominator solving the second degree equation to get:
`f(x)=6/((x+2)(x-1))`
This helps you to "see" the forbidden points, i.e., the points where the denominator becomes zero (you do not want this!!!).
They are:
`x=-2`
`x=1`
Excluding these two values of `x` the other are all allowed.
You can now try to figure out the shape of your graph:
1) for x very big positively or negatively your function gets very small or, better, tends to zero (try to substitute in your function, say, `x=1000` or `x=-1000` you'll find `f(x)~0`);
2) getting near to -2 your function gets very big positively (from the left) and negatively (from the right). You can try it by substituting `-1.999` (on the right of `x=-2`) that gives you `f(x)~-2000` and (on the left of `x=-2`) `x=-2.001` giving `f(x)=2000`. This tendency (for `f(x)` to become very big) is repeated as well when you get near `x=1` (try it!);
3) setting `x=0` gives you `y=-3` which is the y-axis intercept;
At the end your graph looks like:

graph{6/(x^2+x-2) [-10, 10, -5, 5]}

hope it helps