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Simplify the expression:
f(g(x)) = x + 3 + 2\sqrt{x+3} + 1 = x + 2\sqrt{x+3} + 4
Now find \( f(g(5)) \):
g(5) = \sqrt{5+3} = \sqrt{8} = 2\sqrt{2}
Substitute into \( f(x) \):
f(g(5)) = (2\sqrt{2})^2 + 2(2\sqrt{2}) + 1 = 8 + 4\sqrt{2} + 1 = 9 + 4\sqrt{2}
\boxed{9 + 4\sqrt{2}}
Question:** A philosopher of science is analyzing a relationship where \( a + b = 10 \) and \( a^2 + b^2 = 58 \). Determine \( a^3 + b^3 \).
Start by using the identity for the sum of cubes:
a^3 + b^3 = (a+b)(a^2 - ab + b^2)