Question

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

Answer 1

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]}