Question 4 of 5

Select all the correct locations on the image.

What are the domain and the range of this function?

[tex]\[ f(x) = \frac{3}{x+8} - 7 \][/tex]

\begin{tabular}{|l|l|l|}
\hline
[tex]$(y \mid y \in \mathbb{R}, y \neq -7)$[/tex] & [tex]$(y \mid -7 \ \textless \ y \ \textless \ \infty)$[/tex] & [tex]$(y \mid y \in \mathbb{R}, y \neq -5)$[/tex] \\
\hline
[tex]$(x \mid -5 \ \textless \ x \ \textless \ \infty)$[/tex] & [tex]$\{x \mid x \in \mathbb{R}, x \neq -5\}$[/tex] & [tex]$\{x \mid x \in \mathbb{R}, x \neq -7\}$[/tex] \\
\hline
\end{tabular}

Submit



Answer :

To determine the domain and range of the function [tex]\( f(x) = \frac{3}{x+8} - 7 \)[/tex], let’s break it down step-by-step.

1. Finding the Domain:
The domain of a function includes all the possible input values (x-values) for which the function is defined. The function [tex]\( \frac{3}{x+8} - 7 \)[/tex] has a denominator [tex]\( x + 8 \)[/tex]. For the function to be defined, the denominator cannot be zero. Therefore, we set:
[tex]\[ x + 8 \neq 0 \implies x \neq -8 \][/tex]
So, the domain of the function is:
[tex]\[ x \in R, \quad x \neq -8 \][/tex]

2. Finding the Range:
The range of a function includes all the possible output values (y-values). To find the range, we analyze the behavior of the function [tex]\( y = \frac{3}{x+8} - 7 \)[/tex].

First, we examine the part [tex]\( y = \frac{3}{x+8} \)[/tex]. As [tex]\( x \)[/tex] approaches infinity or negative infinity, [tex]\( \frac{3}{x+8} \)[/tex] approaches zero. Consequently:
[tex]\[ f(x) = 0 - 7 = -7 \][/tex]
Therefore, as [tex]\( x \to \infty \)[/tex] or [tex]\( x \to -\infty \)[/tex], the function approaches [tex]\( y = -7 \)[/tex]. However, [tex]\( y = -7 \)[/tex] is never actually reached because [tex]\( \frac{3}{x+8} \)[/tex] never equals zero exactly. This makes [tex]\( y = -7 \)[/tex] a horizontal asymptote.

For other values of [tex]\( x \)[/tex], [tex]\( \frac{3}{x+8} \)[/tex] can take any value except zero, making [tex]\( f(x) \)[/tex] take any value except [tex]\( -7 \)[/tex].

Hence, the range of the function is:
[tex]\[ y \in R, \quad y \neq -7 \][/tex]

From our analysis, we select the locations in the table that represent these findings:

For the domain:
[tex]\[ \{x \mid x \in R, x \neq -8\} \][/tex]

For the range:
[tex]\[ (y \mid y \in R, y \neq -7) \][/tex]

Thus, the correct marked locations are:
1. [tex]\((y \mid y \in R, y \neq -7)\)[/tex]
2. [tex]\(\{x \mid x \in R, x \neq -8\}\)[/tex]