Composite Functions

Composite Functions

Definition and Understanding

  • A composite function is a function that involves two or more functions.
  • It is formed when the output of one function becomes the input of another function.
  • A composite function is denoted as (fog)(x) or f(g(x)), meaning the function f is applied to the output of function g.

Creating Composite Functions

  • To create a composite function, you apply one function to the output of another.
  • Example: if we have two functions, f(x) = x + 3 and g(x) = x^2, their composite function f(g(x)) would be (x^2) + 3.

Evaluating Composite Functions

  • To evaluate a composite function, you start from the innermost function and work outwards.
  • Example: to evaluate f(g(x)) at x = 2, firstly apply g(x) to get g(2) = 2^2= 4, then apply f(x) on the output, f(4) = 4 + 3 = 7. Hence f(g(2))= 7.

Inverse of Composite Functions

  • The inverse of a composite function can be found by reversing the order of operations and replacing each function by its inverse, if it exists.
  • Example: If f(g(x)) is a composite function and both f^-1(x) and g^-1(x) are existing inverses, the inverse of the composite function would be (g^-1of^-1)(x).