To create the table of values for the function \( f(x) = (x + 2)^2 + 1 \) evaluated at \( x = 0 \), \( x = -1 \), and \( x = -2 \), we first need to calculate \( f(x) \) for those values. Afterward, we will compute \( -f(-x) \) for the same \( x \) values.
Step 1: Calculate \( f(x) \)
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For \( x = 0 \): \[ f(0) = (0 + 2)^2 + 1 = 2^2 + 1 = 4 + 1 = 5 \]
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For \( x = -1 \): \[ f(-1) = (-1 + 2)^2 + 1 = 1^2 + 1 = 1 + 1 = 2 \]
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For \( x = -2 \): \[ f(-2) = (-2 + 2)^2 + 1 = 0^2 + 1 = 0 + 1 = 1 \]
Step 2: Calculate \( -f(-x) \)
We also need to evaluate \( f(-x) \) for the same input values:
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For \( x = 0 \): \[ f(-0) = f(0) = 5 \quad \Rightarrow \quad -f(-0) = -5 \]
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For \( x = -1 \): \[ f(-(-1)) = f(1) = (1 + 2)^2 + 1 = 3^2 + 1 = 9 + 1 = 10 \quad \Rightarrow \quad -f(-(-1)) = -10 \]
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For \( x = -2 \): \[ f(-(-2)) = f(2) = (2 + 2)^2 + 1 = 4^2 + 1 = 16 + 1 = 17 \quad \Rightarrow \quad -f(-(-2)) = -17 \]
Step 3: Create the Table
Now we can summarize our findings in the following table:
| \( x \) | \( f(x) \) | \( -f(-x) \) | |----------|------------|--------------| | 0 | 5 | -5 | | -1 | 2 | -10 | | -2 | 1 | -17 |
This table reflects the values of the function \( f(x) \) as well as the negative values of the function evaluated at the negative of those inputs.
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