To determine which statement best describes the function [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex] regarding the value -6, we need to thoroughly examine both the domain and the range of the function.
Step 1: Determine the domain of [tex]\( f(x) \)[/tex].
The domain of the function [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex] is defined by the values of [tex]\( x \)[/tex] for which the expression under the square root is non-negative. This means:
[tex]\[ x - 7 \geq 0 \][/tex]
[tex]\[ x \geq 7 \][/tex]
Thus, the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex].
Step 2: Determine if -6 is in the domain.
Since the domain of [tex]\( f(x) \)[/tex] is [tex]\( x \geq 7 \)[/tex], the number -6 is not in the domain because [tex]\( -6 \)[/tex] does not satisfy [tex]\( x \geq 7 \)[/tex].
Step 3: Determine the range of [tex]\( f(x) \)[/tex].
To find the range, we need to consider the behavior of [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex].
- As [tex]\( x \)[/tex] starts from 7 (the minimum value in the domain), [tex]\( \sqrt{x-7} \)[/tex] starts from 0:
[tex]\[
f(7) = -2 \sqrt{7-7} + 1 = -2(0) + 1 = 1
\][/tex]
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x-7} \)[/tex] becomes larger, making [tex]\( -2 \sqrt{x-7} \)[/tex] more negative. Thus, [tex]\( f(x) \)[/tex] decreases from 1.
Considering the function behaves monotonically decreasings from 1 as [tex]\( x \)[/tex] tends to infinity:
[tex]\[
\text{As} \ x \to \infty, \ \sqrt{x-7} \to \infty \ \Rightarrow \ -2 \sqrt{x-7} \to -\infty, \ \Rightarrow \ f(x) = -2 \sqrt{x-7} + 1 \to -\infty
\][/tex]
Thus, the range of [tex]\( f(x) \)[/tex] is all values [tex]\( y \leq 1 \)[/tex], essentially [tex]\( (-\infty, 1] \)[/tex].
Step 4: Determine if -6 is in the range.
Since the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 1] \)[/tex], -6 is indeed within this range because [tex]\(-6 \leq 1\)[/tex].
Conclusion:
Based on our analysis:
- -6 is not in the domain of [tex]\( f(x) \)[/tex].
- -6 is in the range of [tex]\( f(x) \)[/tex].
The best statement that describes [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex] regarding -6 is:
- [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex] but is in the range of [tex]\( f(x) \)[/tex].
