Composite functions

Introduction to Composite Functions

  • A composite function is a function that is composed of two or more functions, such that the output of one function becomes the input of the next.
  • For two functions, f(x) and g(x), the composite function f(g(x)) is read as ‘f of g of x’, and means that the output of the function g(x) is used as the input for the function f(x).

Defining Composite Functions

  • Composite functions are denoted in the form f(g(x)) or (f o g)(x), both meaning the same thing.
  • The term inside the parentheses (g(x)) is the inner function and the term outside f( ) is the outer function.

Calculating Composite Functions

  • To find the value of a composite function like f(g(x)) for a specific x, first find the value of g(x) (the inner function) for that x.
  • Once the value of g(x) is obtained, substitute it into f(x) (the outer function). This outputs the value of the composite function at that x.

Importance of Order

  • The order of operations in composite functions is significant, changing the order can lead to different results. That is, f(g(x)) is not necessarily equal to g(f(x)).
  • Ensure to work from inside to out: evaluate the inner function first, then feed the result to the outer function.

Examples of Composite Functions

  • For instance, consider two functions f(x) = 2x + 3 and g(x) = x^2. The composite function f(g(x)) is equal to f(x^2) which simplifies to 2x^2 + 3.
  • Conversely, the composite function g(f(x)) is equal to (2x + 3)^2.

Applications of Composite Functions

  • Composite functions are especially useful in real-life applications where one variable is dependent on multiple factors.
  • They can be used to model complex systems in physics, economics, engineering, etc.

Revision Tips

  • Practice is important to solidify understanding of the composite function concept. Try substituting various function forms inside other functions.
  • Work systematically through the process, evaluating the inner function before the outer function.
  • Remember to pay close attention to the order - often, f(g(x)) is not equal to g(f(x)).
  • Draw up tables of values as a visual aid to understanding composite functions.