a) g(f(x)) = sqrt(4 - f(x)) = sqrt(4 - |x-1|).
Domain: 4 - |x-1| must be >=0, i.e. |x-1| <= 4.For x>=1 this means x-1<=4 or x<=5, or x in [1, 5].For x<=1 this means 1-x<=4 or x>=-3, or x in [-3, 1].The entire domain is therefore [-3, 5].
b) f(g(x)) = |g(x)-1| = |sqrt(4-x) - 1|.Here the restriction is 4-x>=0, or x<=4, or x in (-∞, 4].
c) f(f(x)) = |f(x) - 1| = ||x-1|-1|.The domain of this function is (-∞, +∞).
And sorry, I believe that these expressions could not be simplified.
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