Final answer:
To evaluate f(g(2)), we find g(2) = 2 and substitute it into f(x) = 2x(x + 1) to get f(g(2)) = 12. The composite function f(g(x)) is found by substituting g(x) = 5x - 8 into f(x) = 2x(x + 1) to get f(g(x)) = 2(5x - 8)(5x - 7). The composite function g(f(x)) is found by substituting f(x) = 2x(x + 1) into g(x) = 5x - 8 to get g(f(x)) = 10x^2 + 10x - 8.
Step-by-step explanation:
(a) Evaluate f(g(2)):
To evaluate f(g(2)), we need to first find g(2) and then substitute that value into f(x).
Given g(x) = 5x - 8, substituting x = 2, we get: g(2) = 5(2) - 8 = 10 - 8 = 2.
Now, substituting g(2) = 2 into f(x) = 2x(x + 1), we get f(g(2)) = 2(2)((2 + 1) = 2(2)(3) = 12.
(b) Find the composite function f(g(x)):
To find the composite function f(g(x)), we substitute g(x) into f(x).
Substituting g(x) = 5x - 8 into f(x) = 2x(x + 1), we get: f(g(x)) = 2(5x - 8)((5x - 8) + 1) = 2(5x - 8)(5x - 7).
(c) Find the composite function g(f(x)):
To find the composite function g(f(x)), we substitute f(x) into g(x).
Substituting f(x) = 2x(x + 1) into g(x) = 5x - 8, we get: g(f(x)) = 5(2x(x + 1)) - 8 = 10x(x + 1) - 8 = 10x^2 + 10x - 8.