Composite Functions is the application of one function to the results of another function. Composition functions will not be commutative. This commutatively is attained only by particular functions and in special circumstances. Derivatives of compositions that involve differentiable functions can be found using the chain rule. Suppose result of 1st function f (x) is applied into the 2nd function g (x), this composite function is written as g(f(x)) or (g?f) (x). Composite functions are combination of 2 or more functions. We can say that it is a function that is expressed in terms of one or more functions. It may be noted that, (f?g)(x) ≠ (g?f)(x).
Example 1: Simplify by Composite function f(x) = 2x + 3
Solution: First we will apply f and then apply f to that result
=> (f?f)(x) = f(f(x))
=> (f ? f) (x) = 2 (2x + 3) + 3 = 4x + 9
=> f ( f (x) ) = f (2x + 3) = 2 (2x + 3) + 3 = 4x + 9
Example 2: If f(x) = 2x and g(x) = 2^x, then what is (f ? g) (x)?
Solution: The given problem we have
=> F(x) = 2x and g (x) = 2^x
=> When we do composite function (f ? g) (x) we get
=> (f ? g) (x) = f ( g ( x ) ) = f (2^x)
=> And f (2^x) = 2 . 2^x = 2^ (x + 1)
=> Answer: (f ? g) (x) = 2^ (x + 1)