Question
Sun March 18, 2012 By: Sandeep
 

consider functions f and g such that composite gof is defined as one-one.Are f and g both necessarly one-one

Expert Reply
Tue March 20, 2012
Let A be the pro">gof) is a one-to-one function"

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)

e.g.
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.
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