What is function?

A function is a mathematical relationship between two sets of variables, where each input value (independent variable) corresponds to exactly one output value (dependent variable). In simpler terms, a function is a rule or formula that takes an input and produces an output. Functions are commonly represented by equations, graphs, or tables.

Definition:

A function is a mathematical relation between a set of input values and a set of output values, such that for each input value, there is only one corresponding output value.

Key Characteristics:

- Domain: The set of all possible input values.
- Range: The set of all possible output values.
- One-to-One: In a one-to-one function, each input value corresponds to a unique output value.
- Onto: In an onto function, each output value corresponds to at least one input value.

Notation:

A function can be denoted in several ways:

- f(x): The value of the function at input x.
- y = f(x): The output value y is equal to the function of input x.
- f: X → Y, where X is the domain and Y is the range.

Types of Functions:

- Linear: Functions with a constant rate of change.
- Quadratic: Functions with a parabolic shape.
- Exponential: Functions that grow or decay rapidly.
- Trigonometric: Functions related to angles and triangles.

Applications:

Functions are widely used in various fields, including:

- Calculus: Limits, derivatives, and integrals.
- Physics: Describing motion, forces, and energy.
- Economics: Modeling supply and demand.
- Computer Science: Algorithms, data structures, and programming.

Example:

Consider the function f(x) = x^2.

- Domain: All real numbers
- Range: [0, ∞)
- One-to-one: Not one-to-one because f(-x) = f(x).
- Onto: Not onto because the negative output values are not in the range.