Answer:
[tex]\large \boxed{\text{C) }\{y: y \in \mathbb{R}, y \ne 3\} }[/tex]
Step-by-step explanation:
The range is the spread of the y-values (minimum to maximum distance travelled).
The graph of your function is a hyperbola shifted two units left and three units up from the origin.
There is a vertical asymptote at x = -2, so y does not exist when x = -2. However,
[tex]\displaystyle \lim_{x \rightarrow -{2}^{+}}f(x) = \lim_{x \rightarrow -{2}^{+}}\left (\dfrac{1}{x+2}+3 \right ) = 0 + 3 = 3\\\\\lim_{x \rightarrow -{2}^{-}}f(x) = \lim_{x \rightarrow -{2}^{-}}\left (\dfrac{1}{x+2}+3 \right ) = 0 + 3 = 3[/tex]
Since the limit from either side is the same,
[tex]\displaystyle \lim_{x \rightarrow -{2}}f(x) = 3[/tex]
The graph below shows the asymptotes of your function.
Thus. y can take any value except 3.
In set builder notation, the range is
[tex]\large \boxed{\mathbf{\{y: y \in \mathbb{R}, y \ne 3\} }}[/tex]
