# How do you graph f(x)=2/(x-1) using holes, vertical and horizontal asymptotes, x and y intercepts?

Oct 14, 2017

No holes.
Vertical asymptote: $x = 1$
Horizontal asymptote: $y = 0$
No $x$ intercepts.
$y$-intercept: $- 2$

#### Explanation:

Denote $f \left(x\right)$ as (n(x))/(d(x)

There are no holes since there are no common factors.

To find the vertical asymptote,
Solve $d \left(x\right) = 0$
$\Rightarrow$$x - 1 = 0$
$x = 1$

Therefore the vertical asymptote is $x = 1$.

To find the horizontal asymptote,
Compare the leading degree of $n \left(x\right)$ and $d \left(x\right)$.

For $n \left(x\right)$, the degree is $0$, because ${x}^{0} \cdot 2$ gives $2$. Denote this as $\textcolor{t u r q u o i s e}{n}$
For $d \left(x\right)$, the degree is $1$ (since ${x}^{1}$). Denote this as $\textcolor{m a \ge n t a}{m}$

When $n < m$, the $x$-axis (that is, $y = 0$) is the horizontal asymptote.

To find the $x$ intercept, plug in $0$ for $y$ and solve for $x$.
$\Rightarrow$$0 = \frac{2}{x - 1}$
There are no $x$ intercepts.

To find the $y$ intercept, plug in $0$ of $x$ and solve for $y$.
$\Rightarrow$$f \left(x\right) = \frac{2}{0 - 1}$
$f \left(x\right) = - 2$
The $y$-intercept is $- 2$.

graph{2/(x-1 [-100, 100, -5, 5]}