Question 1022638
I will assume the function is {{{ f(x) = (1/2)(x-2)^3 +3 }}}

This function is one-to-one in its domain, hence its inverse function will exist.

Now let {{{ y = (1/2)(x-2)^3 +3 }}}

==> {{{ y-3= (1/2)(x-2)^3 }}}

==> {{{2y-6 = (x-2)^3 }}}

==> {{{root(3, 2y-6) = x-2}}} after taking cube roots of both sides.

==> {{{root(3, 2y-6) + 2 = x}}}

Now interchange the positions of x and y:

 {{{root(3, 2x-6) + 2 = y}}}

This is the inverse function of {{{ f(x) = (1/2)(x-2)^3 +3 }}}