To determine the domain and range of the function [tex]\( f(x)=\frac{3}{x+5}-7 \)[/tex], let's analyze each step thoroughly.
### Domain of the Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For [tex]\( f(x) = \frac{3}{x+5} - 7 \)[/tex], the function is undefined when the denominator is equal to zero. Setting the denominator equal to zero gives:
[tex]\[ x + 5 = 0 \][/tex]
[tex]\[ x = -5 \][/tex]
Therefore, the function [tex]\( f(x) \)[/tex] is undefined at [tex]\( x = -5 \)[/tex]. As a result, the domain of [tex]\( f(x) \)[/tex] includes all real numbers except [tex]\( x = -5 \)[/tex]:
[tex]\[ \{x \mid x \in R, x \neq -5\} \][/tex]
### Range of the Function
The range of a function is the set of all possible output values (y-values).
To find the range, consider the expression [tex]\( f(x) = \frac{3}{x+5} - 7 \)[/tex]. We need to determine which y-values cannot be attained.
Rearranging the equation to express [tex]\( \frac{3}{x+5} \)[/tex]:
[tex]\[ y = \frac{3}{x+5} - 7 \][/tex]
[tex]\[ y + 7 = \frac{3}{x+5} \][/tex]
Notice that [tex]\( \frac{3}{x+5} \)[/tex] can assume any real value except zero because a value of zero in this context would require [tex]\( y + 7 \)[/tex] to be zero (which would allow us to isolate [tex]\( y \)[/tex] to see which values are possible):
[tex]\[ \frac{3}{x+5} = 0 \rightarrow y + 7 = 0 \rightarrow y = -7 \][/tex]
Thus, the function can obtain all real values except [tex]\( y = -7 \)[/tex]. Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ \{y \mid y \in R, y \neq -7\} \][/tex]
### Summary
The domain and range of the function [tex]\( f(x) = \frac{3}{x+5} - 7 \)[/tex] are:
- Domain: [tex]\( \{x \mid x \in R, x \neq -5\} \)[/tex]
- Range: [tex]\( \{y \mid y \in R, y \neq -7\} \)[/tex]
From the given table, the correct choices are:
- Domain: [tex]\( \{x \mid x \in R, x \neq -5\} \)[/tex]
- Range: [tex]\( \{y \mid y \in R, y \neq -7\} \)[/tex]
So, the selections you should make are:
[tex]\[
\{y \mid y \in R, y \neq -7\} \quad \text{and} \quad \{x \mid x \in R, x \neq -5\}
\][/tex]
