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Consider the functions f: R → R given by f(x) = 2x - 1 and g: R {1} → R given by g(x) = (x - 1/2)/(x - 1). The composite function f(g(x)) is:

1. surjective but not injective
2. both injective and surjective
3. injective but not surjective
4. neither injective nor surjective

Correct answer: injective but not surjective

Solution

f(g(x)) = 2*((x-1/2)/(x-1)) - 1 = (2x-1-(x-1))/(x-1) = x/(x-1). This Mobius map is one-to-one (injective) but its range is R\{1}, so it is not surjective onto R: injective but not surjective.

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