# What is an example of a function that has a vertical asymptote at x = -1 and a horizontal asymptote at y = 2?

Mar 5, 2018

$f \left(x\right) \left(= y\right) = 2 + \frac{1}{x + 1}$

#### Explanation:

We know that the relation $\hat{y} = \frac{1}{\hat{x}}$ has a vertical asymptote at $\hat{x} = 0$ and a horizontal asymptote at $\hat{y} = 0$

If we wish to shift all points left one unit (i.e. the vertical asymptote to $x = - 1$) then we need to replace $\hat{x}$ with $x + 1$ (so when $x = - 1$ then $\hat{x} = 0$, the vertical asymptote).

Similarly to shift all points up two units (i.e. the horizontal asymptote to $y = 2$) we need to replace $\hat{y}$ with $y - 2$

Therefore the required relation would become
$\textcolor{w h i t e}{\text{XXX}} y - 2 = \frac{1}{x + 1}$

Converting this into function form (with $y = f \left(x\right)$)
we have
$\textcolor{w h i t e}{\text{XXX}} y = 2 + \frac{1}{x + 1}$