Based on the definition of a composite function, find the composite of f and g.

Question

What is f composite g, written f ο g = f(g(x)) for f(x) = 3x + 4x² -2 and g(x) = x³ ?

Answer 1

To understand composite functions, it's sometimes useful to consider the basic function notation. We define f(x) as a rule that tells you how to manipulate the variable "x". For instance f(x) = x² defines the function "f" as taking the input "x" and squaring it. g(r) = 1/r defines the function g as taking 1 divided by the input "r".

So, then f(g(x)) would be defined as the function f with "g(x)" as the input.

Since f(x) = 3x + 4x² - 2 then f(q) would be 3q + 4q² - 2 and f(?) would be 3(?) + 4(?)² - 2

In this case, since g(x) = x³, then f(g(x)) = 3(x³) + 4(x³)² - 2 which can be simplified as 3x³ + 4x⁶ - 2 or, writing it in standard form, f ο g = f(g(x)) = 4x⁶ + 3x³ - 2

Answer 2

f(g(x) = 3(x³) + 4(x³)² - 2

= 3x³ + 4x⁶ - 2