Question

question 10 (1 point)
determine the range of the function \\(y = \frac{3}{x + 2}\\).

a) \\(\\{y \in \mathbb{r}, y \
eq 0\\}\\)

b) \\(\\{y \in \mathbb{r}\\}\\)

c) \\(\\{x \in \mathbb{r}\\}\\)

d) \\(\\{x \in \mathbb{r}, x \
eq -2\\}\\)

Explanation:

Analyze the rational function structure

We are given the rational function:

$$y = \frac{3}{x + 2}$$

To find the range, we analyze the possible output values of \(y\).

Solve for x in terms of y

We rearrange the equation to express \(x\) as a function of \(y\):

$$y(x + 2) = 3$$

$$x + 2 = \frac{3}{y}$$

$$x = \frac{3}{y} - 2$$

Determine restrictions on y

From the expression \(x = \frac{3}{y} - 2\), we see that \(x\) is defined for all real numbers except when the denominator is zero.
Therefore, we must have:

$$y eq 0$$

Express the range in set-builder notation

Using Set-Builder Notation, the set of all possible output values \(y\) is written as:

$$\{y \in \mathbb{R} \mid y eq 0\}$$

Note: The image displays a slight rendering artifact where the symbols \(\in\) and \(
eq\) are shown as \(\hat{\mathbb{R}}\) and \(
eq\) (or \(\text{l}^1\)), which corresponds to option A: \(\{y \in \mathbb{R}, y
eq 0\}\).

Answer:

• **A) \(\{y \in \mathbb{R}, y

eq 0\}\) (Correct answer)**

• B) \(\{y \in \mathbb{R}\}\)
• C) \(\{x \in \mathbb{R}\}\)
• D) \(\{x \in \mathbb{R}, x

eq -2\}\)