Give an example of a pair of functions f: A -> A and g: A -> A such that f(g(x)) is .........?

Question

a one to one correspondence , but neither f nor g is a one to one correspondence?

Answer 1

This is not possible. If g(x) is not 1-1, then there exist x and y with x ≠ y but g(x) = g(y). Since g(x) = g(y), f(g(x)) = f(g(y)) and hence f(g(x)) is not 1-1.