Composite Functions

Choose a function f(x) from the first row of buttons, and another functions g(x) from the second row. Then, choose the composition of functions whose graph you want to see. If you want to see what the graphs of the individual functions look like, select either or both of the checkboxes for f(x) and g(x). You can simultaneously view the graphs of two of the functions and one composite function.

Exploration 1:
Is composition commutative? In other words, is it true that
(f g)(x) = (g f)(x) for any two functions f(x) and g(x)?

Exploration 2:
Zero is known as the additive identity, because for any number x, both x + 0 and 0 + x are identical to x. Also, one is known as the multiplicative identity, because for any number x, both x⋅1 and 1⋅x are identical to x. Is there an identity for composition? In other words, is there a function I(x) such that for any function f, both f I and I f are identical to f?