Questions: f(x)=x^3-3 f(x) f^(-1)(x) x y -1 0 1 2

Solution Steps

Given the function \(f(x) = x^3 - 3\), we calculate the \(y\) values corresponding to the \(x\) values in the table:

• If \(x = -1\), then \(f(-1) = (-1)^3 - 3 = -1 - 3 = -4\).
• If \(x = 0\), then \(f(0) = (0)^3 - 3 = 0 - 3 = -3\).
• If \(x = 1\), then \(f(1) = (1)^3 - 3 = 1 - 3 = -2\).
• If \(x = 2\), then \(f(2) = (2)^3 - 3 = 8 - 3 = 5\).

So, the table for \(f(x)\) becomes:

\[ \begin{array}{|c|c|} \hline x & y=f(x) \\ \hline -1 & -4 \\ \hline 0 & -3 \\ \hline 1 & -2 \\ \hline 2 & 5 \\ \hline \end{array} \]

To find the inverse, we switch \(x\) and \(y\) in the equation \(y = x^3 - 3\) and solve for \(y\): \(x = y^3 - 3\) \(x + 3 = y^3\) \(y = \sqrt[3]{x + 3}\) So, \(f^{-1}(x) = \sqrt[3]{x + 3}\).

We use the values of \(y\) from the \(f(x)\) table as the input \(x\) values for \(f^{-1}(x)\).

• If \(x = -4\), then \(f^{-1}(-4) = \sqrt[3]{-4 + 3} = \sqrt[3]{-1} = -1\).
• If \(x = -3\), then \(f^{-1}(-3) = \sqrt[3]{-3 + 3} = \sqrt[3]{0} = 0\).
• If \(x = -2\), then \(f^{-1}(-2) = \sqrt[3]{-2 + 3} = \sqrt[3]{1} = 1\).
• If \(x = 5\), then \(f^{-1}(5) = \sqrt[3]{5 + 3} = \sqrt[3]{8} = 2\).

\[ \begin{array}{|c|c|} \hline x & f^{-1}(x) \\ \hline -4 & -1 \\ \hline -3 & 0 \\ \hline -2 & 1 \\ \hline 5 & 2 \\ \hline \end{array} \]

Final Answer