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]
