To do find the inverse of the function, switch the x and y, then solve for the new y:
y = x^3 + 1
becomes
x = y^3 + 1
subtract 1 from each side
x - 1 = y^3
take the cube root of each side (and switch left and right to get things looking normal again)
y = (x-1)^(1/3)
you're done