What is a Function? Definition and Types

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:
$f:X\to Y$
Where $X$ is the domain, $Y$ is the codomain, and $f(x)$ represents the output when $x$ is the input.

Example:
If f(x)=x2f(x) = x^2f(x)=x2, for $x=2$, the output $f(2)=4$.

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Characteristics of a Function

1. Uniqueness: Each input has only one output.
2. Representation: Functions can be expressed using equations, graphs, or tables.
3. 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: $f(x)=mx+c$
Example: $f(x)=2x+3$.

2. Quadratic Function

A function that forms a parabola when graphed.
Formula: f(x)=ax2+bx+cf(x) = ax^2 + bx + cf(x)=ax2+bx+c
Example: f(x)=x2−4x+4f(x) = x^2 - 4x + 4f(x)=x2−4x+4.

3. Polynomial Function

Functions involving powers of $x$.
Formula: f(x)=anxn+⋯+a1x+a0f(x) = a_nx^n + \dots + a_1x + a_0f(x)=an​xn+⋯+a1​x+a0​
Example: f(x)=3x3−2x2+x−5f(x) = 3x^3 - 2x^2 + x - 5f(x)=3x3−2x2+x−5.

4. Rational Function

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

5. Exponential Function

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

6. Logarithmic Function

The inverse of the exponential function.
Formula: f(x)=log⁡a(x)f(x) = \log_a(x)f(x)=loga​(x), $x>0$.
Example: f(x)=log⁡2(x)f(x) = \log_2(x)f(x)=log2​(x).

7. Trigonometric Functions

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

8. Piecewise Function

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

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

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.