Functions – Definition, Examples, and Types of Functions

Functions are one of the fundamental concepts in mathematics, playing a crucial role in various fields like economics, business, and statistics. This blog provides an in-depth understanding of functions, their types, and numerical examples to help you grasp the concept effectively.

1. What is a Function?

A function is a mathematical relationship where each input (from the domain) is associated with exactly one output (in the range). In simpler terms, a function maps elements from one set to another. A function is the rule (or the formula) of how the values in the Domain are related to the values in Rangthe e.

Mathematical Notation:

If f  is a function from set X to set Y, the mathematical notation is:

Or we can write the relationship between X (input / independent variable) and Y (output / dependent variable as:

Key Points:

• Every input has exactly one output (one-to-one correspondence is the core requirement

• Multiple inputs can have one output (many-to-one correspondence is allowed)

• Every output must have a corresponding input (one-to-many is not allowed)

2. Example of a Function

Consider a function

x	f(x)=2x+3
1	2(1) + 3 = 5
2	2(2) + 3 = 7
3	2(3) + 3 = 9

Here, each input x has a unique output f (x), there is a unique output.

3. Types of Functions

Functions can be classified into various types based on their properties and mappings. Here are the major categories:

Figure 1: Types of Functions

A. Based on Mathematical Expression

Figure 2: Types of Functions Based on Properties

i. Linear Function (Straight Line)

A function of the form f (x) = ax + b where a and b are constants.

🔎 Example: f (x) = ax+b

For x = 2, f(2) = 3(2) + 2 = 8

ii. Quadratic Function (Parabola)

A function of the form f(x) = ax^2 + bx + c where a ≠ 0

🔎 Example: f(x) = x^2 − 4x + 3

For x = 2, f(1) = 4 − (4*2) + 3 = -1

iii. Cubic Function (A curve with two turning points)

A function of the form f(x) = ax^3+bx^2 + cx + d

🔎 Example: f(x) = x^3 − 2x^2 + x − 1

For x = 2, f(x) = 8 − (2*4) + 8 − 1 = 7

B. Based on Mapping

i. One-to-One Function (Injective)

Each element of the domain maps to a unique element of the range.

🔎 Example: f(x) = x + 1 is one-to-one since different inputs yield different outputs.

ii. Onto Function (Surjective)

Every element of the range is an image of at least one element of the domain.

🔎 Example: f(x) = x^3 is onto for real numbers.

iii. One-to-One and Onto (Bijective)

A function that is both one-to-one and onto.

🔎 Example: f(x)=x is bijective.