To determine which table shows the values of \( g(x) = -f(-x) \) as a reflection about the origin, we first need to find the points from the initial function \( f(x) = 2(x + 22) \).
Let's compute the values of \( f(x) \) for various values of \( x \):
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For \( x = 1 \): \[ f(1) = 2(1 + 22) = 2(23) = 46 \]
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For \( x = 2 \): \[ f(2) = 2(2 + 22) = 2(24) = 48 \]
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For \( x = 3 \): \[ f(3) = 2(3 + 22) = 2(25) = 50 \]
Now, we can create a table of \( f(x) \) with these values:
- \( x = 1, f(1) = 46 \)
- \( x = 2, f(2) = 48 \)
- \( x = 3, f(3) = 50 \)
Now, let's determine \( g(x) \): \[ g(x) = -f(-x) \]
We need to calculate \( f(-x) \) for \( x = -1, -2, -3 \):
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For \( x = -1 \): \[ f(-(-1)) = f(1) = 46 \quad \Rightarrow \quad g(-1) = -46 \]
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For \( x = -2 \): \[ f(-(-2)) = f(2) = 48 \quad \Rightarrow \quad g(-2) = -48 \]
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For \( x = -3 \): \[ f(-(-3)) = f(3) = 50 \quad \Rightarrow \quad g(-3) = -50 \]
This gives us the following table of \( g(x) \):
- \( x = -1, g(-1) = -46 \)
- \( x = -2, g(-2) = -48 \)
- \( x = -3, g(-3) = -50 \)
Now, we notice a pattern: the function has been reflected about the origin. The \( g(x) \) values are negative of the corresponding \( f(x) \) values from the same \( x \).
Looking through the choices provided, the correct table that reflects the function \( g(x) = -f(-x) \) results in:
\[ \begin{array}{c|c} x & g(x) \ \hline -1 & -46 \ -2 & -48 \ -3 & -50 \ \end{array} \]
Since these values don't match the options you provided, please recheck the values you entered initially for \( f(x) \), as the \( f(-x) \) values seem to differ or might require calculating additional reference points for more clarification on the options given.