In mathematics, function composition is a way of making a new function from two other functions through a chain-like process.
More specifically, given a function f from X to Y and a function g from Y to Z, then the function "g composed with f", written as g ∘ f, is a function from X to Z (notice how it is usually written in the opposite way to how people would expect it to be).
The value of f given the input x is written as f(x). The value of g ∘ f given the input x is written as (g ∘ f)(x), and is defined as g(f(x)).
As an example. let f be a function which doubles a number (multiplies it by 2), and let g be a function which subtracts 1 from a number. These two functions can be written as:
Here, g composed with f would be the function which doubles a number, and then subtracts 1 from it. That is:
On the other hand, f composed with g would be the function which subtracts 1 from a number, and then doubles it:
Composition of functions can also be generalized to binary relations, where it is sometimes represented using the same symbol (as in ).
Properties[change | change source]
Function composition can be proven to be associative, which means that:
However, function composition is in general not commutative, which means that:
This can be also seen in the first example, where (g ∘ f)(2) = 2*2 - 1 = 3 and (f ∘ g)(2) = 2*(2-1) = 2.
Related pages[change | change source]
References[change | change source]
- ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-28.
- ↑ Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28.
- ↑ "3.4: Composition of Functions". Mathematics LibreTexts. 2020-01-16. Retrieved 2020-08-28.