Composition of Functions in Discrete Mathematics: Combining Functions to Create New Mappings

Learn about function composition, a fundamental concept in discrete mathematics. This guide explains how to compose functions, the notation used for function composition, provides examples illustrating the composition of functions, and explores the properties of function composition.

Composition of Functions in Discrete Mathematics

What is Function Composition?

Function composition is a way to combine two functions to create a new function. If we have a function f that maps elements from set A to set B, and another function g that maps elements from set B to set C, then the composition of f and g (written g ∘ f) maps elements from A to C.

Formula for Function Composition

The composition of functions f and g is defined as:

(g ∘ f)(x) = g(f(x))

To evaluate (g ∘ f)(x), you first apply function f to x, and then you apply function g to the result.

Example 1: Function Composition

(This example, showing the composition of functions f and g, is given in the original text and should be included here. The calculation of (g∘f)(1), (g∘f)(2), and (g∘f)(3) should be shown.)

Example 2: Function Composition with Multiple Functions

(This example involving functions f, g, and h on integers is given in the original text and should be included here. The step-by-step calculations for h∘(g∘f)(n), g∘(f∘h)(n), and f∘(g∘h)(n) should be shown.)

Properties of Function Composition

• Associativity: Function composition is associative: h ∘ (g ∘ f) = (h ∘ g) ∘ f.
• Not Commutative: Function composition is generally not commutative: g ∘ f ≠ f ∘ g.
• One-to-One: If f and g are one-to-one, then g ∘ f is also one-to-one.
• Onto: If f and g are onto, then g ∘ f is also onto.

Conclusion

Function composition is a fundamental concept in mathematics, providing a way to build more complex functions from simpler ones. Understanding its properties is essential for working with functions and analyzing their behavior.