Remember the definition of an inverse function. If y = f(x) takes you from x to y, then the inverse function x = f-1(y) takes you back from y to x.
So let's find [f of g](x), in other words f(g(x)):
f(g(x)) = f([x^1/3 - 5]/2)
= (2[x^1/3 - 5]/2 + 5)^3
Cancel those 2s:
= (x^1/3 - 5 + 5)^3
Now cancel the 5s:
Similarly if we find [g of f](x):
g(f(x)) = ([(2x+5)^3]^1/3 - 5)/2
Cancel the ^3 / ^1/3:
g(f(x)) = (2x+5 - 5)/2
And the 5s:
g(f(x)) = 2x/2
See what's happening? If you start with x, and apply the function f to it, you get f(x). But then if you apply g to that value f(x), you get back the original value x. And the same for g(x) - if you apply thr function f to that, you get back the original x than went in to g.
So the two functions are inverses of each other. If you apply one to x, you can get back the original x by applying the other function to your result.