To determine which set of points lies on the graph of the function [tex]\( f(x) = -\sqrt{x} \)[/tex], we will evaluate the function [tex]\( f(x) \)[/tex] at each [tex]\( x \)[/tex]-coordinate provided in the points and check if the calculated [tex]\( y \)[/tex]-value matches the [tex]\( y \)[/tex]-coordinate of the point.
First, let's understand the function [tex]\( f(x) = -\sqrt{x} \)[/tex]. This function only has real values for [tex]\( x \geq 0 \)[/tex], because the square root of a negative number is not a real number. The output of [tex]\( f(x) \)[/tex] is the negative of the square root of [tex]\( x \)[/tex].
We will proceed by checking each list of points:
### List 1: [tex]\((-9,3), (-4,2), (-1,1)\)[/tex]
1. For the point [tex]\((-9, 3)\)[/tex]:
[tex]\[
f(-9) = -\sqrt{-9} \quad \text{(not a real number)}
\][/tex]
2. For the point [tex]\((-4, 2)\)[/tex]:
[tex]\[
f(-4) = -\sqrt{-4} \quad \text{(not a real number)}
\][/tex]
3. For the point [tex]\((-1, 1)\)[/tex]:
[tex]\[
f(-1) = -\sqrt{-1} \quad \text{(not a real number)}
\][/tex]
None of these points are on the graph because they involve the square root of negative numbers, which are not defined in the real number system.
### List 2: [tex]\((1,1), (4,2), (9,3)\)[/tex]
1. For the point [tex]\((1, 1)\)[/tex]:
[tex]\[
f(1) = -\sqrt{1} = -1 \quad \text{(not equal to 1)}
\][/tex]
2. For the point [tex]\((4, 2)\)[/tex]:
[tex]\[
f(4) = -\sqrt{4} = -2 \quad \text{(not equal to 2)}
\][/tex]
3. For the point [tex]\((9, 3)\)[/tex]:
[tex]\[
f(9) = -\sqrt{9} = -3 \quad \text{(not equal to 3)}
\][/tex]
None of these points are on the graph because the calculated [tex]\( y \)[/tex]-values do not match the [tex]\( y \)[/tex]-coordinates of the points.
### List 3: [tex]\((-9,-3), (-4,-2), (-1,-1)\)[/tex]
1. For the point [tex]\((-9, -3)\)[/tex]:
[tex]\[
f(-9) = -\sqrt{-9} \quad \text{(not a real number)}
\][/tex]
2. For the point [tex]\((-4, -2)\)[/tex]:
[tex]\[
f(-4) = -\sqrt{-4} \quad \text{(not a real number)}
\][/tex]
3. For the point [tex]\((-1, -1)\)[/tex]:
[tex]\[
f(-1) = -\sqrt{-1} \quad \text{(not a real number)}
\][/tex]
None of these points are on the graph for the same reasons as list 1.
### List 4: [tex]\((1,-1), (4,-2), (9,-3)\)[/tex]
1. For the point [tex]\((1, -1)\)[/tex]:
[tex]\[
f(1) = -\sqrt{1} = -1 \quad \text{(equal to -1)}
\][/tex]
2. For the point [tex]\((4, -2)\)[/tex]:
[tex]\[
f(4) = -\sqrt{4} = -2 \quad \text{(equal to -2)}
\][/tex]
3. For the point [tex]\((9, -3)\)[/tex]:
[tex]\[
f(9) = -\sqrt{9} = -3 \quad \text{(equal to -3)}
\][/tex]
All of these points are on the graph because for each point, the [tex]\( y \)[/tex]-value is equal to [tex]\( -\sqrt{x} \)[/tex].
So, the list that contains three points that lie on the graph of the function [tex]\( f(x) = -\sqrt{x} \)[/tex] is:
[tex]\[
(1,-1), (4,-2), (9,-3)
\][/tex]
Thus, the correct answer is the fourth list.
