Question 360463
Solving for x in terms of y,
{{{y = x^3-1}}},
{{{y+1 = x^3}}},
{{{(y+1)^(1/3) = x}}}.
Interchange x and y:
{{{(x+1)^(1/3) = y}}}.  The function y is the inverse function.  Note that {{{y = x^3-1}}}is a 1-1 function, by the horizontal line test, so its inverse exists.