find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{x^{2}-16}{3x^{2}-15x…

find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{x^{2}-16}{3x^{2}-15x - 108}$\n\nanswer attempt 1 out of 2\n\nno vertical asymptotes\n\nno vertical asymptotes\n\none vertical asymptote\n\ntwo vertical asymptotes

find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{x^{2}-16}{3x^{2}-15x - 108}$\n\nanswer attempt 1 out of 2\n\nno vertical asymptotes\n\nno vertical asymptotes\n\none vertical asymptote\n\ntwo vertical asymptotes

Answer

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\neq - 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))