To complete the input-output table for the function \( f(x) = -(x + 1)^2 + 4 \), we will calculate the output \( f(x) \) for each given \( x \) value.
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For \( x = -1 \): \[ f(-1) = -((-1) + 1)^2 + 4 = -(0)^2 + 4 = 0 + 4 = 4 \]
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For \( x = 0 \): \[ f(0) = -((0) + 1)^2 + 4 = -1^2 + 4 = -1 + 4 = 3 \]
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For \( x = 1 \): \[ f(1) = -((1) + 1)^2 + 4 = -2^2 + 4 = -4 + 4 = 0 \]
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For \( x = 2 \): \[ f(2) = -((2) + 1)^2 + 4 = -3^2 + 4 = -9 + 4 = -5 \]
Now, we can complete the input-output table:
\[ \begin{array}{c|c} x & f(x) \ \hline -1 & 4 \ 0 & 3 \ 1 & 0 \ 2 & -5 \ \end{array} \]
From the table, we see that the largest output occurs at \( x = -1 \) with \( f(-1) = 4 \).