Question

function
hyperbola:
$f(x)=\frac{1}{x}$
left end behavior (use limit notation):
horizontal asymptote(s):
(write in the form of y = __ )
sketch of graph
right end behavior (use limit notation):
vertical asymptote(s):
(write in the form of x = __ )
write using interval notation:
intervals of increasing:
intervals of decreasing:
write using interval notation:
domain:
write using interval notation:
range:

Explanation:

Step1: Analyze domain

The function $f(x)=\frac{1}{x}$ is undefined at $x = 0$. So the domain is $(-\infty,0)\cup(0,\infty)$.

Step2: Analyze range

As $x$ approaches $0$ from the left or right, $y$ approaches $\pm\infty$. As $x$ approaches $\pm\infty$, $y$ approaches $0$. So the range is $(-\infty,0)\cup(0,\infty)$.

Step3: Find asymptotes

For vertical asymptote, the function is undefined at $x = 0$, so $x=0$ is the vertical asymptote. For horizontal asymptote, $\lim_{x
ightarrow\pm\infty}\frac{1}{x}=0$, so $y = 0$ is the horizontal asymptote.

Step4: Analyze increasing and decreasing intervals

Take the derivative $f^\prime(x)=-\frac{1}{x^{2}}$. Since $f^\prime(x)<0$ for all $x
eq0$, the function is decreasing on $(-\infty,0)$ and $(0,\infty)$ and has no increasing intervals.

Step5: Analyze end - behavior

Left - end behavior: $\lim_{x
ightarrow-\infty}\frac{1}{x}=0$. Right - end behavior: $\lim_{x
ightarrow\infty}\frac{1}{x}=0$.

Answer:

Sketch of Graph: Hyperbola with two branches in opposite quadrants separated by asymptotes.
Intervals of Increasing: None
Intervals of Decreasing: $(-\infty,0)\cup(0,\infty)$
Left End Behavior: $\lim_{x
ightarrow-\infty}\frac{1}{x}=0$
Right End Behavior: $\lim_{x
ightarrow\infty}\frac{1}{x}=0$
Domain: $(-\infty,0)\cup(0,\infty)$
Horizontal Asymptote: $y = 0$
Vertical Asymptote: $x=0$
Range: $(-\infty,0)\cup(0,\infty)$