Question

find all vertical asymptotes of the following function.

$f(x)=\frac{x^{2}-16}{3x^{2}-15x - 108}$

answer attempt 1 out of 2

no vertical asymptotes

no vertical asymptotes

one vertical asymptote

two vertical asymptotes

Explanation:

Step1: Factor numerator and denominator

• Numerator: \(x^{2}-16=(x + 4)(x - 4)\) (using \(a^{2}-b^{2}=(a + b)(a - b)\))
• Denominator: \(3x^{2}-15x - 108=3(x^{2}-5x - 36)=3(x - 9)(x+4)\) (factoring quadratic \(x^{2}-5x - 36=(x - 9)(x + 4)\))

Step2: Simplify the function

\(f(x)=\frac{(x + 4)(x - 4)}{3(x - 9)(x + 4)}=\frac{x - 4}{3(x - 9)},x
eq - 4\)

Step3: Find vertical asymptotes

Vertical asymptotes occur where the denominator of the simplified function is zero (and numerator is non - zero). Set \(3(x - 9)=0\), solve for \(x\). We get \(x = 9\)

Answer:

One Vertical Asymptote (\(x = 9\))