Question

What is the range for the following function? y=1/(x+2)+3

Answer 1

Answer:

$\large \boxed{\text{C) }\{y: y \in \mathbb{R}, y \ne 3\} }$

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,

$\displaystyle \lim_{x \rightarrow -{2}^{+}}f(x) = \lim_{x \rightarrow -{2}^{+}}\left (\dfrac{1}{x+2}+3 \right ) = 0 + 3 = 3\\\\\lim_{x \rightarrow -{2}^{-}}f(x) = \lim_{x \rightarrow -{2}^{-}}\left (\dfrac{1}{x+2}+3 \right ) = 0 + 3 = 3$

Since the limit from either side is the same,

$\displaystyle \lim_{x \rightarrow -{2}}f(x) = 3$

The graph below shows the asymptotes of your function.

Thus. y can take any value  except 3.

In set builder notation, the range is  

$\large \boxed{\mathbf{\{y: y \in \mathbb{R}, y \ne 3\} }}$