Bot GPT 4-o mini To show that \( h(x) \) is the reflection about the origin of the function \( f(x) \), we need to understand how the reflection about the origin works. For a function \( f(x) \), the reflection about the origin \( h(x) \) is defined as:
\[ h(x) = -f(-x) \]
This means we need to evaluate \( f \) at \(-x\) and then multiply that result by \(-1\).
Let's compute \( h(x) \) for each value in the provided table.
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For \( x = 0 \): \[ h(0) = -f(-0) = -f(0) = -0 = 0 \]
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For \( x = 1 \): \[ h(1) = -f(-1) \] Since we don’t have \( f(-1) \) in the provided values, we cannot compute it for \( x = 1 \).
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For \( x = 2 \): \[ h(2) = -f(-2) \] Again, we don’t have \( f(-2) \) in the provided values, so we cannot compute it for \( x = 2 \) either.
Since we only have the values of \( f(x) \) for \( x = 0, 1, \) and \( 2 \) and we do not have their corresponding negative arguments, we cannot fill in the table completely without more information about the function \( f(x) \) for negative values.
If you have any additional information about \( f(x) \) for \( x = -1 \) and \( x = -2 \), please share, and we can continue! But based on the existing data:
The completed table is:
\[ \begin{array}{|c|c|} \hline x & h(x) \ \hline 0 & 0 \ 1 & ? \ 2 & ? \ \hline \end{array} \]
Where \( ? \) indicates missing values we cannot compute without additional information.
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