graph the following rational function.\n f(x)=\frac{3 x}{x - 4} \nfind the nonzontal asymptote(s). select…

graph the following rational function.\n f(x)=\frac{3 x}{x - 4} \nfind the nonzontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the equation(s) of the horizontal asymptote(s) is/are \n(type an equation. use a comma to separate answers as needed.)\nb. there is no horizontal asymptote.\nplot points between and beyond each ( x )-intercept and vertical asymptote. find the value of the function at the given value of ( x )\n f(x)=\frac{3 x}{x - 4} \n(simplify your answer. type an integer or a simplified fraction.)\nuse the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.

graph the following rational function.\n f(x)=\frac{3 x}{x - 4} \nfind the nonzontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the equation(s) of the horizontal asymptote(s) is/are \n(type an equation. use a comma to separate answers as needed.)\nb. there is no horizontal asymptote.\nplot points between and beyond each ( x )-intercept and vertical asymptote. find the value of the function at the given value of ( x )\n f(x)=\frac{3 x}{x - 4} \n(simplify your answer. type an integer or a simplified fraction.)\nuse the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below.

Answer

Explanation:

Step1: Determine the horizontal asymptote

For a rational function (f(x)=\frac{a_nx^n +\cdots}{b_mx^m+\cdots}), if (n = m) (where (n) is the degree of the numerator and (m) is the degree of the denominator), the horizontal asymptote is (y=\frac{a_n}{b_m}). For (f(x)=\frac{3x}{x - 4}), the degree of the numerator (n = 1) and the degree of the denominator (m=1). Here (a_n = 3) and (b_m=1). So the horizontal asymptote is (y=\frac{3}{1}=3).

Step2: Calculate the function values at given (x) - values

  • When (x=-2): (f(-2)=\frac{3\times(-2)}{-2 - 4}=\frac{-6}{-6}=1)
  • When (x = - 1): (f(-1)=\frac{3\times(-1)}{-1-4}=\frac{-3}{-5}=\frac{3}{5})
  • When (x = 3): (f(3)=\frac{3\times3}{3 - 4}=\frac{9}{-1}=-9)
  • When (x = 6): (f(6)=\frac{3\times6}{6 - 4}=\frac{18}{2}=9)
  • When (x = 8): (f(8)=\frac{3\times8}{8 - 4}=\frac{24}{4}=6)

Answer:

  • For the horizontal asymptote: A. The equation(s) of the horizontal asymptote(s) is/are (y = 3)
  • For (f(x)=\frac{3x}{x - 4}) at (x=-2): (1); at (x=-1): (\frac{3}{5}); at (x = 3): (-9); at (x = 6): (9); at (x = 8): (6)