To complete the input-output table for the function \( f(x) = - (x + 1)^2 + 4 \), we will calculate the output for each of the given \( x \) values: -1, 0, 1, and 2.
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For \( x = -1 \): \[ f(-1) = -(-1 + 1)^2 + 4 = -0^2 + 4 = 4 \]
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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 summarize our findings in the table:
\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -1 & 4 \ 0 & 3 \ 1 & 0 \ 2 & -5 \ \hline \end{array} \]
The largest output from our calculated values is \( 4 \), which occurs at \( x = -1 \).
Thus, the answer is: x = -1.