To determine which ordered pairs [tex]\((x, y)\)[/tex] lie on the graph of the function [tex]\(f(x) = -\sqrt[3]{x}\)[/tex], we need to evaluate [tex]\(-\sqrt[3]{x}\)[/tex] for each [tex]\(x\)[/tex] value and see if it matches the corresponding [tex]\(y\)[/tex] value in each pair.
Given function:
[tex]\[ f(x) = -\sqrt[3]{x} \][/tex]
Let's examine each point:
1. For [tex]\((-2, 2)\)[/tex]:
[tex]\[ f(-2) = -\sqrt[3]{-2} \][/tex]
Since the cube root of [tex]\(-2\)[/tex] is [tex]\(-\sqrt[3]{-2} = -(-1.2599) = 1.2599\)[/tex],
[tex]\[ f(-2) \approx -(-1.2599) = 1.2599 \neq 2 \][/tex]
Hence, [tex]\((-2, 2)\)[/tex] is not on the graph.
2. For [tex]\((-1, -1)\)[/tex]:
[tex]\[ f(-1) = -\sqrt[3]{-1} \][/tex]
The cube root of [tex]\(-1\)[/tex] is [tex]\(-1\)[/tex], so:
[tex]\[ f(-1) = -(-1) = 1 \neq -1 \][/tex]
Hence, [tex]\((-1, -1)\)[/tex] is not on the graph.
3. For [tex]\((0, 0)\)[/tex]:
[tex]\[ f(0) = -\sqrt[3]{0} \][/tex]
The cube root of 0 is 0, so:
[tex]\[ f(0) = -0 = 0 \][/tex]
Hence, [tex]\((0, 0)\)[/tex] is on the graph.
4. For [tex]\((1, -1)\)[/tex]:
[tex]\[ f(1) = -\sqrt[3]{1} \][/tex]
The cube root of 1 is 1, so:
[tex]\[ f(1) = -1 = -1 \][/tex]
Hence, [tex]\((1, -1)\)[/tex] is on the graph.
5. For [tex]\((6, -2)\)[/tex]:
[tex]\[ f(6) = -\sqrt[3]{6} \][/tex]
The cube root of 6 is approximately 1.82, so:
[tex]\[ f(6) \approx -1.82 \neq -2 \][/tex]
Hence, [tex]\((6, -2)\)[/tex] is not on the graph.
6. For [tex]\((8, -2)\)[/tex]:
[tex]\[ f(8) = -\sqrt[3]{8} \][/tex]
The cube root of 8 is 2, so:
[tex]\[ f(8) = -2 \][/tex]
Hence, [tex]\((8, -2)\)[/tex] is on the graph.
Thus, the ordered pairs that represent points on the graph of [tex]\(f(x) = -\sqrt[3]{x}\)[/tex] are:
[tex]\[ (0, 0), (1, -1), \text{and} (8, -2). \][/tex]
