We want to draw the asymptotes and the graph of the given function. We will start by considering some possible transformations.

Transformations of $f (x) = \frac{1}{x}, x \neq = 0$
Vertical Translations	$Translation up k units, k > 0 y = \frac{1}{x} + k$
Vertical Translations	$Translation down k units, k > 0 y = \frac{1}{x} - k$
Horizontal Translations	$Translation right h units, h > 0 y = \frac{1}{x - h}$
Horizontal Translations	$Translation left h units, h > 0 y = \frac{1}{x + h}$

Note that if the graph of the function is translated, the asymptotes are also translated in the same distance and direction. Consider the function.

y = \frac{1}{x - 2} + 5

The given function is a combination of transformations.

Let's apply these transformations one at a time. We will start by translating the parent function, $f (x) = \frac{1}{x},$ $2$ units right.

The last transformation is a vertical translation of $5$ units up.

Finally, let's look at the graph of the given function and its asymptotes alone.

We can see that the vertical asymtpote is the line

x = 2,

and the equation of the horizontal asymptote is

y = 5 .

Using this information, we can state the domain and range of the function.

Domain : Range : All real numbers except x = 2 All real numbers except y = 5

Exercise