find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{5 x^{2}-12 x-9}{2 x^{2}-16…

find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{5 x^{2}-12 x-9}{2 x^{2}-16 x+30}$\n\nno vertical asymptotes\none vertical asymptote\ntwo vertical asymptotes\nno vertical asymptotes

find all vertical asymptotes of the following function.\n\n$f(x)=\\frac{5 x^{2}-12 x-9}{2 x^{2}-16 x+30}$\n\nno vertical asymptotes\none vertical asymptote\ntwo vertical asymptotes\nno vertical asymptotes

Answer

Explanation:

Step1: Factor numerator and denominator

  • Numerator: (5x^{2}-12x - 9=(5x + 3)(x-3))
  • Denominator: (2x^{2}-16x + 30 = 2(x^{2}-8x + 15)=2(x - 3)(x - 5))

Step2: Simplify the function

(f(x)=\frac{(5x + 3)(x - 3)}{2(x - 3)(x - 5)}=\frac{5x+3}{2(x - 5)},x\neq3)

Step3: Find vertical asymptotes

Vertical asymptotes occur where the denominator is zero (after simplification). Set (2(x - 5)=0), solve for (x). We get (x = 5)

Answer:

One Vertical Asymptote