Question 51040
If {{{f(x) = (x-2)^(1/3)}}}, find {{{f^-1(x)}}}
First, replace f(x) with y, so now you have:
{{{y = (x-2)^(1/3)}}} Cube both sides.
{{{y^3 = x-2}}} Interchange the variables.
{{{x^3 = y-2}}} Now solve for y.
{{{y = x^3+2}}} Finally, replace y with {{{f^-1(x)}}}
{{{f^-1(x) = x^3+2}}} This is choice f. in your list.

Check:
{{{f(f^-1(x)) = f(x^3+2)}}} = {{{((x^3+2)-2)^(1/3) = (x^3)^(1/3)}}} = x
{{{f^-1(f(x)) = f^-1(x-2)^(1/3)}}} = {{{((x^3+2)-2)^(1/3) = (x^3)^(1/3)}}} = x

The above check is based on the following:
f(x) and g(x) are inverses of each other iff:
f(g(x)) = x for all x in the domain of g, and g(f(x)) = x for all x in the domain of f.