Answer :
To determine which statement best describes [tex]\( f(x) = -2\sqrt{x - 7} + 1 \)[/tex] with respect to the value [tex]\(-6\)[/tex], we need to analyze both the domain and the range of the function, and then check if [tex]\(-6\)[/tex] fits these descriptions.
### Step 1: Determine the Domain of [tex]\( f(x) \)[/tex]
For [tex]\( f(x) \)[/tex] to be defined:
[tex]\[ x - 7 \geq 0 \][/tex]
[tex]\[ x \geq 7 \][/tex]
The domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
### Step 2: Determine if [tex]\(-6\)[/tex] is in the Domain of [tex]\( f(x) \)[/tex]
Since the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex], [tex]\(-6 \)[/tex] is not in the domain because [tex]\(-6 < 7\)[/tex].
### Step 3: Determine the Range of [tex]\( f(x) \)[/tex]
Let's find the range of the function [tex]\( f(x) \)[/tex].
1. Since [tex]\(x \geq 7\)[/tex], we start with [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = -2\sqrt{7 - 7} + 1 = -2\sqrt{0} + 1 = 1 \][/tex]
2. As [tex]\( x \)[/tex] increases, [tex]\( x - 7 \)[/tex] becomes positive and [tex]\(\sqrt{x - 7}\)[/tex] increases. Therefore, [tex]\( -2 \sqrt{x - 7} \)[/tex] becomes more negative.
For large [tex]\( x \)[/tex], [tex]\( -2\sqrt{x - 7} \)[/tex] can be very negative. So:
[tex]\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left( -2\sqrt{x - 7} + 1 \right) = -\infty \][/tex]
Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ (-\infty, 1] \][/tex]
### Step 4: Determine if [tex]\(-6\)[/tex] is in the Range of [tex]\( f(x) \)[/tex]
Given the range is [tex]\((- \infty, 1]\)[/tex], [tex]\(-6 \)[/tex] falls within this range because [tex]\(-\infty < -6 \leq 1\)[/tex].
### Conclusion
Based on the analysis:
- [tex]\(-6 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
- [tex]\(-6 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
Thus, the best statement that describes the function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] with respect to [tex]\(-6\)[/tex] is:
"-6 is not in the domain of [tex]\( f(x) \)[/tex] but is in the range of [tex]\( f(x) \)[/tex]".
### Step 1: Determine the Domain of [tex]\( f(x) \)[/tex]
For [tex]\( f(x) \)[/tex] to be defined:
[tex]\[ x - 7 \geq 0 \][/tex]
[tex]\[ x \geq 7 \][/tex]
The domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
### Step 2: Determine if [tex]\(-6\)[/tex] is in the Domain of [tex]\( f(x) \)[/tex]
Since the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex], [tex]\(-6 \)[/tex] is not in the domain because [tex]\(-6 < 7\)[/tex].
### Step 3: Determine the Range of [tex]\( f(x) \)[/tex]
Let's find the range of the function [tex]\( f(x) \)[/tex].
1. Since [tex]\(x \geq 7\)[/tex], we start with [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = -2\sqrt{7 - 7} + 1 = -2\sqrt{0} + 1 = 1 \][/tex]
2. As [tex]\( x \)[/tex] increases, [tex]\( x - 7 \)[/tex] becomes positive and [tex]\(\sqrt{x - 7}\)[/tex] increases. Therefore, [tex]\( -2 \sqrt{x - 7} \)[/tex] becomes more negative.
For large [tex]\( x \)[/tex], [tex]\( -2\sqrt{x - 7} \)[/tex] can be very negative. So:
[tex]\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \left( -2\sqrt{x - 7} + 1 \right) = -\infty \][/tex]
Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ (-\infty, 1] \][/tex]
### Step 4: Determine if [tex]\(-6\)[/tex] is in the Range of [tex]\( f(x) \)[/tex]
Given the range is [tex]\((- \infty, 1]\)[/tex], [tex]\(-6 \)[/tex] falls within this range because [tex]\(-\infty < -6 \leq 1\)[/tex].
### Conclusion
Based on the analysis:
- [tex]\(-6 \)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
- [tex]\(-6 \)[/tex] is within the range of [tex]\( f(x) \)[/tex].
Thus, the best statement that describes the function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] with respect to [tex]\(-6\)[/tex] is:
"-6 is not in the domain of [tex]\( f(x) \)[/tex] but is in the range of [tex]\( f(x) \)[/tex]".
