Functions in Math

Functions

• A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range), assigning each input to exactly one output.

Notation

• Functions are denoted by letters such as f, g, and h.
• The notation f(x) indicates the output of the function f when the input is x.

Types of Functions

One-to-One (Injective) Function

• A function is one-to-one if each output is associated with only one input.

Onto (Surjective) Function

• A function is onto if every output is associated with at least one input.

Bijective Function

• A function is bijective if it is both one-to-one and onto.

Function Operations

Composition

• The composition of two functions f and g, denoted as (f ∘ g)(x), applies g to the input and then applies f to the output.

Inverse

• The inverse of a function f, denoted as f^(-1), reverses the output and input of f.

Graphical Representation

• The graph of a function visually represents the relationship between the input and output.
• The graph determines the domain and range of the function.

Properties of Functions

Domain

• The set of inputs for which the function is defined.

Range

• The set of possible outputs of the function.

Identity

• The identity function, denoted as I, maps each input to itself.

Constant

• A constant function maps every input to a single output.

Function Examples

Linear Function

• A function of the form f(x) = mx + b, where m and b are constants.

Quadratic Function

• A function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

Exponential Function

• A function of the form f(x) = a^x, where a is a constant.

Polynomial Function

• A function of the form f(x) = a_n x^n + a_(n-1) x^(n-1) +...+ a_1 x + a_0, where a_n,..., a_0 are constants.