To complete the input-output table for the function \( f(x) = - (x + 1)^2 + 4 \), we can calculate \( f(x) \) for the given values of \( x \):
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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 the output in a table:
\[ \begin{array}{c|c} x & f(x) \ \hline -1 & 4 \ 0 & 3 \ 1 & 0 \ 2 & -5 \ \end{array} \]
The outputs are:
- \( f(-1) = 4 \)
- \( f(0) = 3 \)
- \( f(1) = 0 \)
- \( f(2) = -5 \)
The largest output occurs at \( x = -1 \) with \( f(-1) = 4 \).
Thus, the correct response to the question is:
\( x = -1 \), but it was not provided as an option. However, if we must choose among the given options, none correctly produce the largest output, which is \( 4 \) at \( x = -1 \). If limited to responses, \( x = 0 \) produces the next largest output of \( 3 \).