Bot GPT 4-o mini To find the horizontal reflection of the function \( f(x) = x^3 \), we need to compute \( -f(x) \). This means we will take the function values at the given \( x \) values and multiply them by -1.
First, we'll compute \( f(x) = x^3 \) for each \( x \) value and then take the negative of that value to get \( -f(x) \).
Here are the calculations:
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For \( x = -3 \): \[ f(-3) = (-3)^3 = -27 \quad \Rightarrow \quad -f(-3) = -(-27) = 27 \]
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For \( x = -2 \): \[ f(-2) = (-2)^3 = -8 \quad \Rightarrow \quad -f(-2) = -(-8) = 8 \]
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For \( x = -1 \): \[ f(-1) = (-1)^3 = -1 \quad \Rightarrow \quad -f(-1) = -(-1) = 1 \]
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For \( x = 0 \): \[ f(0) = 0^3 = 0 \quad \Rightarrow \quad -f(0) = -0 = 0 \]
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For \( x = 1 \): \[ f(1) = 1^3 = 1 \quad \Rightarrow \quad -f(1) = -1 \]
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For \( x = 2 \): \[ f(2) = 2^3 = 8 \quad \Rightarrow \quad -f(2) = -8 \]
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For \( x = 3 \): \[ f(3) = 3^3 = 27 \quad \Rightarrow \quad -f(3) = -27 \]
Now we can fill in the table:
\[ \begin{array}{|c|c|} \hline x & -f(x) \ \hline -3 & 27 \ -2 & 8 \ -1 & 1 \ 0 & 0 \ 1 & -1 \ 2 & -8 \ 3 & -27 \ \hline \end{array} \]
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