To find which \( x \) value produces the largest output for the function \( f(x) = -(x^2) + 4 \), we'll calculate \( f(x) \) for each given \( x \) value: -1, 1, 0, and 2.
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For \( x = -1 \): \[ f(-1) = -((-1)^2) + 4 = -1 + 4 = 3 \]
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For \( x = 1 \): \[ f(1) = -(1^2) + 4 = -1 + 4 = 3 \]
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For \( x = 0 \): \[ f(0) = -(0^2) + 4 = -0 + 4 = 4 \]
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For \( x = 2 \): \[ f(2) = -(2^2) + 4 = -4 + 4 = 0 \]
Now, let's summarize the outputs:
- \( f(-1) = 3 \)
- \( f(1) = 3 \)
- \( f(0) = 4 \)
- \( f(2) = 0 \)
The largest output is \( 4 \), which occurs at \( x = 0 \).
Thus, the \( x \) value that produces the largest output is \( \boxed{0} \).