To determine which statement best describes the function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex], we need to carefully analyze both the domain and the range of [tex]\( f(x) \)[/tex]. Let's go through this step-by-step:
1. Domain of [tex]\( f(x) \)[/tex]:
- The function involves a square root, [tex]\( \sqrt{x - 7} \)[/tex]. The expression inside the square root, [tex]\( x - 7 \)[/tex], must be non-negative because the square root of a negative number is not defined in the set of real numbers.
- Therefore, we need [tex]\( x - 7 \geq 0 \)[/tex], which simplifies to [tex]\( x \geq 7 \)[/tex].
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( [7, \infty) \)[/tex].
Now we check whether [tex]\(-6\)[/tex] is in this domain:
- Clearly, [tex]\(-6 < 7\)[/tex], so [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
2. Range of [tex]\( f(x) \)[/tex]:
- To find the range, we consider the output values of the function. The square root function [tex]\( \sqrt{x - 7} \)[/tex] yields non-negative values (i.e., [tex]\( \sqrt{x - 7} \geq 0 \)[/tex] for [tex]\( x \geq 7 \)[/tex]).
- Thus, [tex]\( -2 \sqrt{x - 7} \)[/tex] yields non-positive values (i.e., [tex]\( -2 \sqrt{x - 7} \leq 0 \)[/tex]).
- Adding 1 shifts these values, so [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] will yield values that are less than or equal to 1.
- When [tex]\( x = 7 \)[/tex], [tex]\( f(x) = -2 \cdot 0 + 1 = 1 \)[/tex].
- As [tex]\( x \)[/tex] increases indefinitely, [tex]\( \sqrt{x - 7} \)[/tex] becomes larger and larger, making [tex]\( -2 \sqrt{x - 7} \)[/tex] tend towards negative infinity.
- Thus, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 1] \)[/tex].
Now we check whether [tex]\(-6\)[/tex] is in this range:
- Since [tex]\(-6 \leq 1\)[/tex], [tex]\(-6\)[/tex] is indeed in the range of [tex]\( f(x) \)[/tex].
3. Conclusion:
- From our analysis, we have determined that [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex] but it is in the range of [tex]\( f(x) \)[/tex].
Hence, the statement that best describes the function [tex]\( f(x) = -2 \sqrt{x - 7} + 1 \)[/tex] in relation 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]."
