Question

let $f(x) = \frac{x^2 + x - 6}{x^2 + 3x - 18}$. enter the equations of all vertical asymptotes of $f(x)$. separate multiple equations with a comma and if there are no vertical asymptotes, enter none.

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 + x - 6\): \(x^2 + x - 6=(x + 3)(x - 2)\)
Factor \(x^2 + 3x - 18\): \(x^2 + 3x - 18=(x + 6)(x - 3)\)
So \(f(x)=\frac{(x + 3)(x - 2)}{(x + 6)(x - 3)}\)

Step2: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (and numerator is not zero at those points).
Set denominator \((x + 6)(x - 3)=0\).
Solve: \(x + 6 = 0\) gives \(x=-6\); \(x - 3 = 0\) gives \(x = 3\).
Check numerator at \(x=-6\): \((-6 + 3)(-6 - 2)=(-3)(-8)=24
eq0\).
Check numerator at \(x = 3\): \((3 + 3)(3 - 2)=(6)(1)=6
eq0\).

Answer:

\(x=-6, x = 3\)