which statement best describes $f(x) = -2\\sqrt{x - 7} + 1$?\n- $-6$ is in the domain of $f(x)$ but not in…

which statement best describes $f(x) = -2\\sqrt{x - 7} + 1$?\n- $-6$ is in the domain of $f(x)$ but not in the range of $f(x)$.\n- $-6$ is not in the domain of $f(x)$ but is in the range of $f(x)$.\n- $-6$ is in the domain of $f(x)$ and in the range of $f(x)$.\n- $-6$ is neither in the domain of $f(x)$ nor in the range of $f(x)$.

which statement best describes $f(x) = -2\\sqrt{x - 7} + 1$?\n- $-6$ is in the domain of $f(x)$ but not in the range of $f(x)$.\n- $-6$ is not in the domain of $f(x)$ but is in the range of $f(x)$.\n- $-6$ is in the domain of $f(x)$ and in the range of $f(x)$.\n- $-6$ is neither in the domain of $f(x)$ nor in the range of $f(x)$.

Answer

Explanation:

Step1: Check Domain

For the function ( f(x) = -2\sqrt{x - 7} + 1 ), the expression under the square root must be non - negative. So we set up the inequality ( x - 7\geq0 ), which gives ( x\geq7 ). Since ( - 6<7 ), ( -6 ) is not in the domain of ( f(x) ).

Step2: Check Range

Let's find if ( -6 ) can be in the range. We set ( f(x)=-6 ), so ( -6=-2\sqrt{x - 7}+1 ). First, we isolate the square root term: Subtract 1 from both sides: ( -6 - 1=-2\sqrt{x - 7} ), so ( -7=-2\sqrt{x - 7} ). Then divide both sides by - 2: ( \frac{-7}{-2}=\sqrt{x - 7} ), or ( \sqrt{x - 7}=\frac{7}{2} ). Squaring both sides, we get ( x - 7=\left(\frac{7}{2}\right)^2=\frac{49}{4} ), and ( x=\frac{49}{4}+7=\frac{49 + 28}{4}=\frac{77}{4}=19.25\geq7 ). So when ( x = \frac{77}{4} ), ( f(x)=-6 ), which means ( -6 ) is in the range of ( f(x) ).

Answer:

-6 is not in the domain of ( f(x) ) but is in the range of ( f(x) ) (the option corresponding to this statement among the given options)