Question

find all vertical asymptotes of the following function.

$f(x)=\frac{x^{2}-49}{2x^{2}+18x + 28}$

Explanation:

Step1: Factor numerator and denominator

Numerator: \(x^{2}-49=(x + 7)(x - 7)\)
Denominator: \(2x^{2}+18x + 28=2(x^{2}+9x + 14)=2(x + 2)(x+7)\)
So \(f(x)=\frac{(x + 7)(x - 7)}{2(x + 2)(x + 7)}\)

Step2: Simplify the function

Cancel out the common factor \((x + 7)\) (for \(x
eq-7\)), we get \(f(x)=\frac{x - 7}{2(x + 2)}\), \(x
eq-7\)

Step3: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (after simplification). Set \(2(x + 2)=0\), solve for \(x\)
\(x+2 = 0\Rightarrow x=-2\)

Answer:

\(x=-2\)