Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Experience the ease of finding reliable answers to your questions from a vast community of knowledgeable experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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]
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]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.