What is a Function? Definition and Types

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What is a Function? Definition and Types

A function is a fundamental concept in mathematics that explains a relationship between two sets of values: the input (domain) and the output (codomain). Each input corresponds to exactly one output, making functions a precise way to describe how one quantity depends on another.

Definition of a Function

A function is a rule that assigns each element of a set (domain) to a unique element in another set (codomain). Mathematically, a function $f$ is written as:
$\to Y$
Where $X$ is the domain, $Y$ is the codomain, and $f (x)$ represents the output when $x$ is the input.

Example:
If $f(x) = x^2$ , for $x = 2$ , the output $f (2) = 4$ .

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Characteristics of a Function

Uniqueness: Each input has only one output.
Representation: Functions can be expressed using equations, graphs, or tables.
Dependency: Outputs depend directly on inputs, creating a cause-effect relationship.

Types of Functions

Functions come in various forms based on their properties and applications. Here are the primary types:

1. Linear Function

A function where the graph is a straight line.
Formula: $f (x) = m x + c$
Example: $f (x) = 2 x + 3$ .

2. Quadratic Function

A function that forms a parabola when graphed.
Formula: $f(x) = ax^2 + bx + c$
Example: $f(x) = x^2 - 4x + 4$ .

3. Polynomial Function

Functions involving powers of $x$ .
Formula: $a_nx^n + \dots + a_1x + a_0$
Example: $f(x) = 3x^3 - 2x^2 + x - 5$ .

4. Rational Function

A ratio of two polynomial functions.
Formula: $\frac{P(x)}{Q(x)}$ , where $\neq 0$ .
Example: $\frac{x+1}{x-2}$ .

5. Exponential Function

A function where the variable appears as an exponent.
Formula: $f(x) = a^x$ , $a > 0$ , $\neq 1$ .
Example: $f(x) = 2^x$ .

6. Logarithmic Function

The inverse of the exponential function.
Formula: $f(x) = \log_a(x)$ , $x > 0$ .
Example: $f(x) = \log_2(x)$ .

7. Trigonometric Functions

Functions based on the angles of a triangle.
Examples: $\sin(x)$ , $\cos(x)$ .

8. Piecewise Function

A function defined by different rules for different intervals of the domain.
Example:

$x≤0\begin{cases} x + 1, & \text{if } x > 0 \\ x - 1, & \text{if } x \leq 0 \end{cases}$

Functions are powerful tools for modelling real-world problems and understanding relationships between variables. Whether in physics, engineering, or economics, they provide a structured way to describe change and dependency.