Question

find all asymptotes, x - intercepts, and y - intercepts for the graph of the rational function and sketch the graph of the function.
f(x)=\frac{1}{x - 5}
find all vertical asymptotes. select the correct choice below and, if necessary, fill in the answer box within your choice.
a. x = 5 (simplify your answer. use a comma to separate answers as needed.)
b. there are no vertical asymptotes.
find the horizontal or oblique asymptote. select the correct choice below and, if necessary, fill in the answer box within your choice.
a. y=
(simplify your answer.)
b. there is no horizontal nor oblique asymptote.

Explanation:

Step1: Find horizontal asymptote

For a rational function $f(x)=\frac{1}{x - 5}$, the degree of the numerator $n = 0$ and the degree of the denominator $m=1$. When $n

Step2: Find x - intercept

Set $y = f(x)=0$, so $\frac{1}{x - 5}=0$. There is no solution for this equation since the numerator is 1 (non - zero), so there are no x - intercepts.

Step3: Find y - intercept

Set $x = 0$, then $f(0)=\frac{1}{0 - 5}=-\frac{1}{5}$.

Answer:

Horizontal asymptote: A. $y = 0$
x - intercepts: There are no x - intercepts
y - intercept: $y=-\frac{1}{5}$