Proves that Let f : A B and g : B C be functions.

A) Prove that If f has a well-defined inverse (bijective), g f must also have a well-defined inverse(bijective). (this is the actually problem and anything below is my own thought and I don't know if is right or not?)

How to prove this to be a false statement, giving a counter-example

Let A = {1, 2} and B = {3, 4} and C = {5, 6, 7}

f(1) = 3 and f(2) = 4 proves that f is well defined inverse

g(f(1)) = 5 and g(f(2)) = 6 proves that g o f is not a well-defined inverse because the 7 got left out (it is not onto proves that it is not well define inverse) thus it is a false statement

Is this a valid counter-example for the statement or not?