Question

which statement best describes $f(x) = -2\sqrt{x - 7} + 1$?

• $-6$ is not in the domain of $f(x)$ but is in the range of $f(x)$.
• $-6$ is neither in the domain of $f(x)$ nor in the range of $f(x)$.
• $-6$ is in the domain of $f(x)$ but not in the range of $f(x)$.
• $-6$ is in the domain of $f(x)$ and in the range of $f(x)$.

Explanation:

Step1: Analyze Domain of \( f(x) = -2\sqrt{x - 7}+1 \)

The square root function \( \sqrt{x - 7} \) requires \( x - 7 \geq 0 \), so \( x \geq 7 \). Since \( -6 < 7 \), \( -6 \) is not in the domain.

Step2: Analyze Range of \( f(x) \)

The square root \( \sqrt{x - 7} \geq 0 \), so \( -2\sqrt{x - 7} \leq 0 \), and \( f(x)= -2\sqrt{x - 7}+1 \leq 1 \). Let's check if \( -6 \) can be in the range: set \( -2\sqrt{x - 7}+1 = -6 \), then \( -2\sqrt{x - 7} = -7 \), \( \sqrt{x - 7}=\frac{7}{2} \), \( x - 7=\frac{49}{4} \), \( x=\frac{49}{4}+7=\frac{77}{4}\geq7 \). So \( -6 \) is in the range (since \( x=\frac{77}{4} \) is valid, so \( f(\frac{77}{4})=-6 \)).

Answer:

\(-6\) is not in the domain of \( f(x) \) but is in the range of \( f(x) \). (The option with this statement)