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Sagot :
To determine if the given ordered pairs [tex]\((0, -2)\)[/tex] and [tex]\((-1, -3)\)[/tex] are on the graph of the function [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex], we need to evaluate the function at each specific [tex]\(x\)[/tex]-value and compare the result with the corresponding [tex]\(y\)[/tex]-value.
### Step-by-Step Solution:
1. For the ordered pair [tex]\((0, -2)\)[/tex]:
- Evaluate the function [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex].
- The function is [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex].
- Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = -\left(\frac{1}{2}\right)^0 - 1 \][/tex]
- Calculate the exponent:
[tex]\[ \left(\frac{1}{2}\right)^0 = 1 \][/tex]
- Substitute back into the function:
[tex]\[ f(0) = -1 - 1 = -2 \][/tex]
- Compare with the [tex]\(y\)[/tex]-value in the ordered pair:
[tex]\[ y = -2 \][/tex]
- Conclusion: Since [tex]\(f(0) = -2\)[/tex], the ordered pair [tex]\((0, -2)\)[/tex] is On the graph.
2. For the ordered pair [tex]\((-1, -3)\)[/tex]:
- Evaluate the function [tex]\(f(x)\)[/tex] at [tex]\(x = -1\)[/tex].
- The function is [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex].
- Substitute [tex]\(x = -1\)[/tex]:
[tex]\[ f(-1) = -\left(\frac{1}{2}\right)^{-1} - 1 \][/tex]
- Calculate the exponent:
[tex]\[ \left(\frac{1}{2}\right)^{-1} = 2 \][/tex]
- Substitute back into the function:
[tex]\[ f(-1) = -2 - 1 = -3 \][/tex]
- Compare with the [tex]\(y\)[/tex]-value in the ordered pair:
[tex]\[ y = -3 \][/tex]
- Conclusion: Since [tex]\(f(-1) = -3\)[/tex], the ordered pair [tex]\((-1, -3)\)[/tex] is On the graph.
### Final Answer:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Ordered pair} & \text{On the graph} & \text{Not on the graph} \\ \hline (0, -2) & \text{On the graph} & \\ \hline (-1, -3) & \text{On the graph} & \\ \hline \end{array} \][/tex]
Both ordered pairs [tex]\((0, -2)\)[/tex] and [tex]\((-1, -3)\)[/tex] are on the graph of the function [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex].
### Step-by-Step Solution:
1. For the ordered pair [tex]\((0, -2)\)[/tex]:
- Evaluate the function [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex].
- The function is [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex].
- Substitute [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = -\left(\frac{1}{2}\right)^0 - 1 \][/tex]
- Calculate the exponent:
[tex]\[ \left(\frac{1}{2}\right)^0 = 1 \][/tex]
- Substitute back into the function:
[tex]\[ f(0) = -1 - 1 = -2 \][/tex]
- Compare with the [tex]\(y\)[/tex]-value in the ordered pair:
[tex]\[ y = -2 \][/tex]
- Conclusion: Since [tex]\(f(0) = -2\)[/tex], the ordered pair [tex]\((0, -2)\)[/tex] is On the graph.
2. For the ordered pair [tex]\((-1, -3)\)[/tex]:
- Evaluate the function [tex]\(f(x)\)[/tex] at [tex]\(x = -1\)[/tex].
- The function is [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex].
- Substitute [tex]\(x = -1\)[/tex]:
[tex]\[ f(-1) = -\left(\frac{1}{2}\right)^{-1} - 1 \][/tex]
- Calculate the exponent:
[tex]\[ \left(\frac{1}{2}\right)^{-1} = 2 \][/tex]
- Substitute back into the function:
[tex]\[ f(-1) = -2 - 1 = -3 \][/tex]
- Compare with the [tex]\(y\)[/tex]-value in the ordered pair:
[tex]\[ y = -3 \][/tex]
- Conclusion: Since [tex]\(f(-1) = -3\)[/tex], the ordered pair [tex]\((-1, -3)\)[/tex] is On the graph.
### Final Answer:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Ordered pair} & \text{On the graph} & \text{Not on the graph} \\ \hline (0, -2) & \text{On the graph} & \\ \hline (-1, -3) & \text{On the graph} & \\ \hline \end{array} \][/tex]
Both ordered pairs [tex]\((0, -2)\)[/tex] and [tex]\((-1, -3)\)[/tex] are on the graph of the function [tex]\(f(x) = -\left(\frac{1}{2}\right)^x - 1\)[/tex].
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