Bot GPT 4-o mini To complete the table for the horizontal reflection of the function \( f(x) = x^3 \), we first need to find \( -f(x) \) for each \( x \) value provided. The horizontal reflection of the function is essentially multiplying the original function by -1.
Let's compute \( f(x) = x^3 \) and then find \( -f(x) \):
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For \( x = -3 \): \[ f(-3) = (-3)^3 = -27 \quad \Rightarrow \quad -f(-3) = 27 \]
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For \( x = -2 \): \[ f(-2) = (-2)^3 = -8 \quad \Rightarrow \quad -f(-2) = 8 \]
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For \( x = -1 \): \[ f(-1) = (-1)^3 = -1 \quad \Rightarrow \quad -f(-1) = 1 \]
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For \( x = 0 \): \[ f(0) = 0^3 = 0 \quad \Rightarrow \quad -f(0) = 0 \]
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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, let’s summarize these calculations in the table:
\[ \begin{array}{|c|c|} \hline x & -f(x) \ \hline -3 & 27 \ -2 & 8 \ -1 & 1 \ 0 & 0 \ 1 & -1 \ 2 & -8 \ 3 & -27 \ \hline \end{array} \]
This is the completed table for the horizontal reflection of the function \( f(x) = x^3 \).
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