Question

find all vertical asymptotes of the following function.
$f(x) = \frac{x^2 + 7x + 12}{2x + 6}$
answer attempt 1 out of 2
no vertical asymptotes
no vertical asymptotes
one vertical asymptote
two vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

Factor \(x^2 + 7x + 12\) as \((x + 3)(x + 4)\), and factor \(2x + 6\) as \(2(x + 3)\). So \(f(x)=\frac{(x + 3)(x + 4)}{2(x + 3)}\).

Step2: Simplify the function

Cancel the common factor \((x + 3)\) (note \(x
eq - 3\) to avoid division by zero in the original function). The simplified function is \(f(x)=\frac{x + 4}{2}\) (a linear function), which has no vertical asymptotes (vertical asymptotes occur where the denominator is zero and the numerator is non - zero at that point; here, after simplification, the denominator is a non - zero constant, and the original denominator's zero at \(x=-3\) is a removable discontinuity, not a vertical asymptote).

Answer:

No Vertical Asymptotes