Question

give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function f(x) = \frac{3x + 2}{x - 1}. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the vertical asymptote is x = 1. (type an equation.) b. there is no vertical asymptote. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the horizontal asymptote is y = 3. (type an equation.) b. there is no horizontal asymptote. select the correct answer below and, if necessary, fill in the answer box to complete your choice. a. the oblique asymptote is. (type an equation.) b. there is no oblique asymptote.

Explanation:

Step1: Find vertical asymptote

Set the denominator equal to 0. For $f(x)=\frac{3x + 2}{x - 1}$, we have $x-1=0$, so $x = 1$.

Step2: Find horizontal asymptote

Since the degree of the numerator and denominator are the same (both degree 1), divide the leading - coefficients. The leading coefficient of the numerator is 3 and of the denominator is 1. So $y=\frac{3}{1}=3$.

Step3: Check for oblique asymptote

For a rational function $\frac{f(x)}{g(x)}$, if the degree of $f(x)$ is exactly one more than the degree of $g(x)$, there is an oblique asymptote. Here, the degrees of the numerator and denominator are equal, so there is no oblique asymptote.

Answer:

A. The vertical asymptote is $x = 1$.
A. The horizontal asymptote is $y = 3$.
B. There is no oblique asymptote.