To determine which points lie on the graph of the function [tex]\( f(x) = -\sqrt{-x} \)[/tex], we need to check each given point [tex]\((x, y)\)[/tex] to see if the y-coordinate at that point matches the output of the function when evaluated at the x-coordinate.
Let's analyze each point step by step:
1. Point [tex]\((-4, -2)\)[/tex]:
[tex]\[
x = -4, y = -2
\][/tex]
[tex]\[
f(-4) = -\sqrt{-(-4)} = -\sqrt{4} = -2
\][/tex]
The y-coordinate [tex]\(-2\)[/tex] matches [tex]\(f(-4)\)[/tex]. Therefore, [tex]\((-4, -2)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
2. Point [tex]\((-1, 1)\)[/tex]:
[tex]\[
x = -1, y = 1
\][/tex]
[tex]\[
f(-1) = -\sqrt{-(-1)} = -\sqrt{1} = -1
\][/tex]
The y-coordinate [tex]\(1\)[/tex] does not match [tex]\(f(-1)\)[/tex] which is [tex]\(-1\)[/tex]. Therefore, [tex]\((-1, 1)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
3. Point [tex]\((0, 0)\)[/tex]:
[tex]\[
x = 0, y = 0
\][/tex]
[tex]\[
f(0) = -\sqrt{-0} = -\sqrt{0} = 0
\][/tex]
The y-coordinate [tex]\(0\)[/tex] matches [tex]\(f(0)\)[/tex]. Therefore, [tex]\((0, 0)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
4. Point [tex]\((1, 1)\)[/tex]:
[tex]\[
x = 1, y = 1
\][/tex]
[tex]\[
f(1) = -\sqrt{-(1)}
\][/tex]
Since the argument of the square root becomes negative (i.e., [tex]\(-1\)[/tex]), [tex]\(f(1)\)[/tex] is undefined in the real number system. Therefore, [tex]\((1, 1)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
5. Point [tex]\((2, -4)\)[/tex]:
[tex]\[
x = 2, y = -4
\][/tex]
[tex]\[
f(2) = -\sqrt{-(2)}
\][/tex]
Again, since the argument of the square root is negative (i.e., [tex]\(-2\)[/tex]), [tex]\(f(2)\)[/tex] is undefined in the real number system. Therefore, [tex]\((2, -4)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
6. Point [tex]\((9, 3)\)[/tex]:
[tex]\[
x = 9, y = 3
\][/tex]
[tex]\[
f(9) = -\sqrt{-(9)}
\][/tex]
Since the argument of the square root is negative (i.e., [tex]\(-9\)[/tex]), [tex]\(f(9)\)[/tex] is undefined in the real number system. Therefore, [tex]\((9, 3)\)[/tex] does not lie on the graph of [tex]\(f(x)\)[/tex].
Based on our calculations, the points that lie on the graph of [tex]\( f(x) = -\sqrt{-x} \)[/tex] are:
[tex]\[
(-4, -2) \quad \text{and} \quad (0, 0)
\][/tex]
