Answer :

To find the inverse of the function [tex]\( f(x) = \sqrt[3]{x - 2} \)[/tex], we need to follow these steps:

1. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = \sqrt[3]{x - 2} \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ x = \sqrt[3]{y - 2} \][/tex]

3. Isolate [tex]\( y \)[/tex] by undoing the cube root. To do this, cube both sides of the equation:
[tex]\[ x^3 = y - 2 \][/tex]

4. Solve for [tex]\( y \)[/tex]:
[tex]\[ y = x^3 + 2 \][/tex]

Thus, the inverse function [tex]\( f^{-1}(x) \)[/tex] is:
[tex]\[ f^{-1}(x) = x^3 + 2 \][/tex]

So the inverse of the function [tex]\( f(x) = \sqrt[3]{x - 2} \)[/tex] is:
[tex]\[ f^{-1}(x) = x^3 + 2 \][/tex]