The Lesson
A composite function is a function of a function. A composite function combines two or more functions so that the output of one function becomes the input of another. Imagine a function f that relates an input x to an output f(x). The output is passed into another function g, which relates it to an output gf(x).The composite function gf relates the input x straight to the output gf(x). The composite function is denoted gf(x), (g ∘ f)(x) or g(f(x)).
A Real Example of a Composite Function
It is easier to understand composite functions with an example.f(x) = 2x and g(x) = x + 1
Consider two functions:f(x) = 2x
g(x) = x + 1
- The function f(x) = 2x takes each input and doubles it.
- The function g(x) = x + 1 takes each input and adds 1 to it.
The Composite Function gf(x)
Now let's consider the composite function of f(x) = 2x and g(x) = x + 1.- The function f relates an input x to an output 2x (it doubles the input).
- The output 2x is the input of the function g, which relates it to an output 2x + 1 (it adds 1 to the input).
The composite of f(x) = 2x and g(x) = x + 1 is:
The Order of Composite Functions Matter
In the example above, the output of f(x) became the input of g(x). The composite function was gf(x) = 2x + 1. Consider what happens if the output of g(x) became the input of f(x).- The function g relates an input x to an output x + 1 (it adds 1 to the input).
- The output x + 1 is the input of the function f, which relates it to an output 2(x + 1) (it doubles the input).
The composite of g(x) = x + 1 and f(x) = 2x is:
fg(x) is not the same as gf(x). The order of the functions matter.
gf(x) ≠ fg(x)
A Note on Notation
The image below shows a composite function gf(x), where a function f is passed to a function g: The composite function is gf(x).- f is applied to the input x.
- g is applied to the function f.
- start with the input x
- write the f that is applied to it to its left
- write the g that is applied to it to its left
Composite Functions with More Than Two Functions
A composite function can be made from more than two functions Imagine there are three functions: f(x), g(x) and h(x). If f is passed into g, which is passed into h...
→ f( ) → g( ) → h( ) →
The composite function will be hgf(x) or (h ∘ g ∘ f)(x) or h(g(f(x))).
Composite Function of a Function and its Inverse
If a composite function is made of a function and its inverse function, the output is the input x:
ff^{−1} = f^{−1}f = x