Answer:
g(f(x) = 3x² + 18x + 16
Explanation:
Definition of Composite Function
Given functions f(x) and g(x) , the composite function, g(f(x) is found by substituting f(x) for x in g(x)
In this particular instance we have
f(x) = x + 2
g(x) = 3x² + 6x - 8
g(f(x)) ==> g(x + 2)
For g(x) = 3x² + 6x - 8, substitute x with x + 2
so we get
g(x + 2) = 3(x + 2)² + 6(x + 2) - 8
Computing and simplifying each of the terms:
(x + 2)² = x² + 4x + 4 from (a + b)² = a² + 2ab + b²
3(x + 2)² = 3(x² + 4x + 4)
= 3x² + 12 x + 12
6(x + 2) = 6x + 12
Putting all this together,
3(x + 2)² + 6(x + 2) - 8
= (3x² + 12 x + 12) + (6x + 12) - 8
= 3x² + 12x + 6x + 12 + 12 -8
= 3x² + 18x + 16
∴ g(f(x) = 3x² + 18x + 16 Answer