To determine which list contains three points that lie on the graph of the function [tex]\( f(x) = -\sqrt{x} \)[/tex], we can evaluate the function at the specified [tex]\( x \)[/tex]-values and check the corresponding [tex]\( y \)[/tex]-values.
The function given is [tex]\( f(x) = -\sqrt{x} \)[/tex].
Let's evaluate the function at the given [tex]\( x \)[/tex]-values for each list:
### List 1: [tex]\((-9, 3), (-4, 2), (-1, 1)\)[/tex]
1. Evaluate [tex]\( f(-9) \)[/tex]:
[tex]\[
f(-9) = -\sqrt{-9}
\][/tex]
The square root of a negative number is not real, so [tex]\((-9, 3)\)[/tex] does not lie on the graph.
2. Evaluate [tex]\( f(-4) \)[/tex]:
[tex]\[
f(-4) = -\sqrt{-4}
\][/tex]
Similarly, the square root of a negative number is not real, so [tex]\((-4, 2)\)[/tex] does not lie on the graph.
3. Evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = -\sqrt{-1}
\][/tex]
The square root of a negative number is not real, so [tex]\((-1, 1)\)[/tex] does not lie on the graph.
### List 2: [tex]\((1, 1), (4, 2), (9, 3)\)[/tex]
1. Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = -\sqrt{1} = -1
\][/tex]
So, [tex]\((1, 1)\)[/tex] does not lie on the graph because [tex]\( f(1) = -1 \neq 1 \)[/tex].
2. Evaluate [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = -\sqrt{4} = -2
\][/tex]
So, [tex]\((4, 2)\)[/tex] does not lie on the graph because [tex]\( f(4) = -2 \neq 2 \)[/tex].
3. Evaluate [tex]\( f(9) \)[/tex]:
[tex]\[
f(9) = -\sqrt{9} = -3
\][/tex]
So, [tex]\((9, 3)\)[/tex] does not lie on the graph because [tex]\( f(9) = -3 \neq 3 \)[/tex].
### List 3: [tex]\((-9, -3), (-4, -2), (-1, -1)\)[/tex]
1. Evaluate [tex]\( f(-9) \)[/tex]:
[tex]\[
f(-9) = -\sqrt{-9}
\][/tex]
The square root of a negative number is not real, so [tex]\((-9, -3)\)[/tex] does not lie on the graph.
2. Evaluate [tex]\( f(-4) \)[/tex]:
[tex]\[
f(-4) = -\sqrt{-4}
\][/tex]
The square root of a negative number is not real, so [tex]\((-4, -2)\)[/tex] does not lie on the graph.
3. Evaluate [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = -\sqrt{-1}
\][/tex]
The square root of a negative number is not real, so [tex]\((-1, -1)\)[/tex] does not lie on the graph.
### List 4: [tex]\((1, -1), (4, -2), (9, -3)\)[/tex]
1. Evaluate [tex]\( f(1) \)[/tex]:
[tex]\[
f(1) = -\sqrt{1} = -1
\][/tex]
So, [tex]\( (1, -1) \)[/tex] lies on the graph.
2. Evaluate [tex]\( f(4) \)[/tex]:
[tex]\[
f(4) = -\sqrt{4} = -2
\][/tex]
So, [tex]\( (4, -2) \)[/tex] lies on the graph.
3. Evaluate [tex]\( f(9) \)[/tex]:
[tex]\[
f(9) = -\sqrt{9} = -3
\][/tex]
So, [tex]\( (9, -3) \)[/tex] lies on the graph.
Therefore, the list that contains three points lying on the graph of the function [tex]\( f(x) = -\sqrt{x} \)[/tex] is:
[tex]\[
\boxed{(1, -1), (4, -2), (9, -3)}
\][/tex]
