Question

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

A){y: y∈ℝ, y≠-2}
B){y: y∈ℝ, y≠2}
C){y: y∈ℝ, y≠3}
D){y: y∈ℝ}

Answer 1

`\large \boxed{\text{C) }\{y: y \in \mathbb{R}, y \\e 3\} }`

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 ((1)/(x+2)+3 \right ) = 0 + 3 = 3\\\\\lim_{x \rightarrow -{2}^(-)}f(x) = \lim_{x \rightarrow -{2}^(-)}\left ((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 \\e 3\} }}`