How do you solve this equation? (Composition of Functions)?

f(x) = x² + 2x
g(x) = 3x² ‒ 1

f(g(x))
= f(3x² ‒ 1)
= (3x² ‒ 1)² + 2(3x² ‒ 1)
= 9x⁴ ‒ 6x² + 1 + 6x² ‒ 2
= 9x⁴ ‒ 1

For any composition of functions h = f ∘ g (where h is defined on the domain of f ∘ g), observe that whatever expression we have for g, that becomes the "x" in f(x) So, in your example where f(x) = -4x + 7 and g(x) = 10x - 6 we have f(g(x)) = f(10x - 6) = -4(10x - 6) + 7 = -40x + 24 + 7 = -40x + 31 Note that it does not necessarily follow that f(g(x)) = g(f(x)) For, consider g(f(x)) in your example: g(f(x)) = g(-4x + 7) = 10(-4x + 7) - 6 = -40x + 70 - 6 = -40x + 64 Depending on the nature of the functions, the order of the compositions can result in very different outcomes Whichever way the composition is formulated, just substitute the expression in the first function for x and consider that the "new x"

f(g(x))=f(3x^2-1) put 3x^2-1=y
=f(y)
=y^2+2y
= (x^2-1)^2+2(3x^2-1) putting the value y
=x^4-2x^2+6x-2

f ( 3x^2-1) = (3x^2-1)^2 +2(3x^2-1)
= 9x^4 -6x^2 +1 +6x^2 -2
= 9x^4 -1

= 9x^4-1