Question

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

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 \(x = 8\): \(6\)