Consider the two functions: f(x) = 1-x and g(x) = 1/x. We can compose them in two ways: f(g(x)) and g(f(x)). We can go further and compose these two new functions with themselves, and also with the old ones, in a number of ways. Keeping composing these functions with the new ones as they are generated and figure out simplified formulae for them in terms of the variable x.
a) How many distinct functions are there, including f and g themselves?
b) List them
c) How is each one composed from f and g?
d) How do you know that these are all there are?
e) For what real numbers are all these functions simultaneously defined?