Determine the inverse function of j(x):
$\begin{array}{l}j(x)=\sqrt[3]{x+3} \\ j^{-1}(x)=\end{array}$

Answer

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Answer

$\boxed{j^{-1}(x) = x^3 - 3}$

Steps

Step 1 ：$y = \sqrt[3]{x+3}$

Step 2 ：$x = \sqrt[3]{y+3}$

Step 3 ：$x^3 = y + 3$

Step 4 ：$y = x^3 - 3$

Step 5 ：$j^{-1}(x) = x^3 - 3$

Step 6 ：$j(j^{-1}(x)) = \sqrt[3]{(x^3 - 3) + 3} = \sqrt[3]{x^3} = x$

Step 7 ：$j^{-1}(j(x)) = (j(x))^3 - 3 = (\sqrt[3]{x+3})^3 - 3 = x + 3 - 3 = x$

Step 8 ：$\boxed{j^{-1}(x) = x^3 - 3}$