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

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

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Answer 1 · 2017-10-14T07:45:21

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

Explanation:

Denote $f(x)$ as $(n(x))/(d(x)$

There are no holes since there are no common factors.

To find the vertical asymptote,
Solve $d(x)=0$
$rArr$ $x-1=0$
$x=1$

Therefore the vertical asymptote is $x=1$ .

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

For $n(x)$ , the degree is $0$ , because $x^0*2$ gives $2$ . Denote this as $color(turquoise)n$
For $d(x)$ , the degree is $1$ (since $x^1$ ). Denote this as $color(magenta)m$

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 $x$ .
$rArr$ $0=2/(x-1)$
There are no $x$ intercepts.

To find the $y$ intercept, plug in $0$ of $x$ and solve for $y$ .
$rArr$ $f(x)=2/(0-1)$
$f(x)=-2$
The $y$ -intercept is $-2$ .

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

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

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