Question 1157859
here's a reference on the properties of functions that are inverse of each other.
<a href = "https://www.analyzemath.com/inversefunction/properties_inverse.html" target = "_blank">https://www.analyzemath.com/inversefunction/properties_inverse.html</a>
the graph of inverses are reflective of each other across the line y = x.
this can be shown in the following graph.
<img src = "http://theo.x10hosting.com/2020/050201.jpg" alt="$$$" >
the red line is the graph of the original equation of y = x^3.
the blue line is the graph of the inverse equation of y = x^(1/3)
the lin y = -x + 4 is there to allow me to show you that the point (x,y) on the graph of the original equation is opposite and equidistant from the line y = x and the reflective point is (y,x) on the graph of the inverse equation.
specifically, the point (1.379,2.621) on the graph of the original equation is equidistant from the line y = x as the point (2.621,1.379) on the inverse equation.


i'm not sure i understand the last part of your question, so i won't try to answer it.
hopefully, what i have provided is helpful to you.
most of the properties of inverse functions are in the reference.


there is another properties of inverse functions that may be what you are alluding to in the last part of your question.
that is that (fog)x) = x if f and g are inverse functions.
in your original example, that is confiemd as shown below.
(fog)(x) = f(g(x)).
(gof(x) = g(f(x)).
if they are inverses of each other, the f(g(x)) = x and g(f(x)) = x.
specifically.
f(g(x)) = (x^(1/3))^3 = x
g(f(x)) = (x^3)^(1/3) = x
check the reference.
lots of good stuff in there.