consider functions f and g such that composite gof is defined as one-one.Are f and g both necessarly one-one
And let B be the pro;
therefore (gof)(x) = g(f(x)) = g(f(y)) = (gof)(y)
therefore (gof) is not one-to-one.
For (gof) to be one-to-one it is only necessary for g to be one-to-one over the range of f. It could do any weird and wonderful things outside that range but that would not affect the properties of the composite function (gof)
Define the functions f and g over the natural numbers such that
f(a) = 2a
g(a) = a if a is even, 1 if a is odd
so (gof)(a) = 2a
(gof) is a one-to-one function even though g is not.