Answer:
We have defined functions:
f(x) = IxI + 1
g(x) = 1/x^3.
Currently, it is evident that the composite functions are not commutative.
How can we demonstrate this?
To determine if two composite functions are commutative, the following must hold true:
f(g(x)) = g(f(x))
One could apply brute force (simply substituting values to see if the composite functions commute),
but I will opt for a more sophisticated approach.
There are two notable observations:
g(x) has a point of discontinuity at x = 0.
Thus:
f(g(x)) = I 1/x^3 I + 1
remains discontinuous at x = 0, whereas:
g(f(x)) = 1/(IxI + 1)^3
shows that the denominator IxI + 1 can never reach zero.
At this point, there is no discontinuity.
Consequently, the composite functions cannot be commutative.
