Let's start by recalling possible transformations of the parent function $f (x) = \frac{1}{x} .$

Function	Transformation of the Graph of $f (x) = \frac{1}{x}$
$g (x) = \frac{1}{x - h}$	Horizontal translation by $h$ units. If $h > 0,$ the translation is to the right. If $h < 0,$ the translation is to the left.
$g (x) = \frac{1}{x} + k$	Vertical translation by $k$ units. If $k > 0,$ the translation is up. If $k < 0,$ the translation is down.
$g (x) = a (\frac{1}{x})$	Vertical stretch or compression by a factor of $a .$ If $a > 1,$ it is a vertical stretch. If $0 < a < 1,$ it is a vertical compression.

Now, let's consider the given function.

g (x) = 3 (\frac{1}{x + 1}) - 2 ⇕ g (x) = 3 (\frac{1}{x - ( - 1 )}) + (- 2)

We can see that

a = 3,

h = - 1,

and

k = - 2 .

From here, we can determine the transformations.

A vertical stretch by a factor of $3 .$
A horizontal translation $1$ unit to the left.
A vertical translation $2$ units down.

Using these transformations, we can find the asymptotes and the reference points of the graph of $g (x) .$ Note that the horizontal translation affects only the $x -$ coordinates, while the vertical stretch and vertical translation affect only the $y -$ coordinates.

Object	$f (x) = \frac{1}{x}$	$g (x) = 3 (\frac{1}{x - ( - 1 )}) + (- 2)$
Vertical asymptote	$x = 0$	$x = 0 + (- 1)$ $⇕$ $x = - 1$
Horizontal asymptote	$y = 0$	$y = 0 + (- 2)$ $⇕$ $y = - 2$
Reference point	$(- 1, - 1)$	$(- 1 + (- 1), 3 (- 1) + (- 2))$ $⇕$ $(- 2, - 5)$
Reference point	$(1, 1)$	$(1 + (- 1), 3 (1) + (- 2))$ $⇕$ $(0, 1)$

Next, we will use the table above to graph $f (x)$ and $g (x) .$

Finally, we will state the domain and range of $g (x) .$ Since $x = - 1$ is the vertical asymptote and $y = - 2$ is the horizontal asymptote, we will exclude them from the domain and range, respectively.

Notation	Domain	Range
Inequality	$x < - 1$ or $x > - 1$	$y < - 2$ or $y > - 2$
Set Notation	${x ∣ x \neq = - 1}$	${y ∣ y \neq = - 2}$
Interval Notation	$(- \infty, - 1) \cup (- 1, + \infty)$	$(- \infty, - 2) \cup (- 2, + \infty)$

Exercise