Question

find all asymptotes, x - intercepts, and y - intercepts for the graph of the rational function and sketch the graph of the function.
$f(x)=\frac{1}{x - 5}$
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. $y = 0$ (simplify your answer.)
b. there is no horizontal nor oblique asymptote.
find the x - intercept(s). select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the x - intercept(s) are (simplify your answer. type an ordered pair. use a comma to separate answers as needed.)
b. there are no x - intercepts.
find the y - intercept. select the correct choice below and, if necessary, fill in the answer box within your choice.
a. the y - intercept is $-\frac{1}{5}$ (simplify your answer. type an ordered pair.)
b. there is no y - intercept.

Explanation:

Step1: Find vertical asymptote

Set denominator equal to 0. For $f(x)=\frac{1}{x - 5}$, $x-5=0$ gives $x = 5$.

Step2: Find horizontal asymptote

Degree of numerator is 0 and degree of denominator is 1. When degree of numerator < degree of denominator, $y = 0$ is the horizontal asymptote.

Step3: Find x - intercept

Set $y=0$, so $\frac{1}{x - 5}=0$. There is no solution for this equation as a non - zero constant divided by a non - zero number (or any number in this case) can't be 0, so no x - intercepts.

Step4: Find y - intercept

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

Answer:

Vertical asymptote: A. $x = 5$
Horizontal or oblique asymptote: A. $y = 0$
x - intercept(s): B. There are no x - intercepts
y - intercept: A. The y - intercept is $(0,-\frac{1}{5})$