Mathematics\Algebra\Functions

Functions are central objects of study in the branch of mathematics known as algebra. At its core, a function is a relation between two sets that associates every element of the first set with exactly one element of the second set.

Definition and Notation

Mathematically, a function is often denoted by \( f: A \rightarrow B \), where:
- \( A \) is called the domain of the function, the set from which inputs are taken.
- \( B \) is the codomain, the set into which outputs are mapped.
- \( f(x) \) represents the output (or the image) of the function corresponding to an input \( x \in A \).

Types of Functions

Functions can be classified in various ways based on their properties:
- Injective functions (or one-to-one): Each element of the codomain is mapped by at most one element of the domain. Formally, \( f \) is injective if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).
- Surjective functions (or onto): Every element of the codomain is the image of at least one element of the domain. This means the range of \( f \), denoted \( \text{range}(f) \), is equal to the codomain \( B \).
- Bijective functions: Functions that are both injective and surjective. They establish a one-to-one correspondence between elements of the domain and codomain, making them invertible.

Mathematical Representation

A function can be represented in various forms:
- Graphically: As a curve or line in a coordinate system, where each input \( x \) is associated with an output \( f(x) \).
- Algebraically: Using an equation, such as \( f(x) = x^2 + 2x + 1 \).
- Tabularly: A table listing input-output pairs.

Example

Consider the quadratic function \( f(x) = x^2 + 2x + 1 \):
1. Domain: All real numbers \( \mathbb{R} \).
2. Codomain: All real numbers \( \mathbb{R} \).
3. Expression: \( f(x) = (x + 1)^2 \), a parabola that opens upwards with vertex at \( (-1,0) \).

Important Concepts

1. Image and Preimage: If \( y = f(x) \), then \( y \) is the image of \( x \), and \( x \) is a preimage of \( y \).
2. Inverse Function: If \( f \) is bijective, the inverse function \( f^{-1} \) maps each element of the codomain back to its corresponding element in the domain.
3. Composite Function: If \( g: B \rightarrow C \) and \( f: A \rightarrow B \), the composite function \( g \circ f: A \rightarrow C \) is defined by \( (g \circ f)(x) = g(f(x)) \).

Functions form the foundation for understanding many other concepts in algebra and analysis, making them indispensable in both pure and applied mathematics.