let $f(x)=\\frac{3x^{2}-16x + 5}{3x^{2}-8x + 4}$\nthis function has (enter answers separated by commas if…

let $f(x)=\\frac{3x^{2}-16x + 5}{3x^{2}-8x + 4}$\nthis function has (enter answers separated by commas if more than one):\n1) vertical asymptotes at $x=$\n2) horizontal asymptote at $y=$\nquestion help: message instructor

let $f(x)=\\frac{3x^{2}-16x + 5}{3x^{2}-8x + 4}$\nthis function has (enter answers separated by commas if more than one):\n1) vertical asymptotes at $x=$\n2) horizontal asymptote at $y=$\nquestion help: message instructor

Answer

Explanation:

Step1: Factor numerator and denominator

Factor (3x^{2}-16x + 5=(3x - 1)(x - 5)) Factor (3x^{2}-8x + 4=(3x - 2)(x - 2)) So (f(x)=\frac{(3x - 1)(x - 5)}{(3x - 2)(x - 2)})

Step2: Find vertical asymptotes

Vertical asymptotes occur where denominator is (0) (and numerator is not (0)). Set (3x-2 = 0), then (x=\frac{2}{3}) Set (x - 2=0), then (x = 2)

Step3: Find horizontal asymptote

Since degree of numerator (n = 2) and degree of denominator (m=2) (where (n=m)) The horizontal asymptote (y=\frac{a_{n}}{b_{m}}), where (a_{n}) is leading coefficient of numerator and (b_{m}) is leading coefficient of denominator. Here (a_{n}=3), (b_{m}=3), so (y = 1)

Answer:

  1. (x=\frac{2}{3},2)
  2. (y = 1)