Question

How do you find the domain for `R(x) = (x^2 + x - 12)/(x^2 - 4)`?

Answer 1

Find that `R(x)` is well defined for all `x` except `+-2`, so the domain of `R(x)` is `(-oo, -2) uu (-2, 2) uu (2, oo)`.

Explanation:

Both the numerator and denominator are well defined for any Real value of `x`. So the quotient `R(x)` will be well defined except when the denominator is zero.

`x^2-4 = 0` when `x = +-sqrt(4) = +-2`

So the domain of `R(x)` is `(-oo, -2) uu (-2, 2) uu (2, oo)`

Note that `x^2+x-12 = (x+4)(x-3)`, so the numerator is non-zero when `x = +-2`. As a result, `R(x)` has vertical asymptotes at `x=+-2` ...

graph{(x^2+x-12)/(x^2-4) [-20, 20, -10, 10]}