Answer
(a) $x^{2}$ + 5, Domain = All Real
(b) $x^{2}$ +2 x + 5, Domain = All Real
(c) x + 2, Domain = All Real
(d) $x^{4}$ + 8$x^{2}$ + 20 , Domain = All Real
Work Step by Step
With f(x) = x + 1, and g(x) = $x^{2}$ + 4
(a) f(g(x)) = f($x^{2}$ + 4) = $x^{2}$ + 4 + 1 = $x^{2}$ + 5
The domain of f(x) is x, and the domain of g(x) is x, so the overall domain = all real
(b) g(f(x)) = g(x + 1) = $(x+1)^{2}$ + 4 = $x^{2}$ +2 x + 5
The domain of f(x) is x, and the domain of g(x) is x, so the overall domain = all real
(c) f(f(x)) = f(x+1) = (x+1+1) = x + 2, domain = all real.
(d) g(g(x)) = g(g(x)) = g($x^{2}$ + 4) = $(x^{2} + 4)^{2}$ + 4 = $x^{4}$ + 8$x^{2}$ + 20
The domain of g(x) is x, so the overall domain = all real
