this is the pp based on functions and types of functions

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1. HELLO EVERYONE! This project is to give information about Functions
and Types of Functions.

2. FUNCTIONS AND TYPES OF FUNCTIONS. By
Somnath Gorai
Aman Kumar
Aditya Kashyap
Sagar Kumar Soni
Amit Kumar Mahto

3. WHAT IS A FUNCTION? A function is a special relationship between two sets where: each input (from the
first set, called domain) has exactly one output (from the second set, called co-
domain).
Symbolically, a function is written as:
f : X → Y
where X = Domain, Y = Co-domain.
Example:
f(x)=2x+3
If input x=2, output = f(2)=7.

4. BASIC CONCEPTS OF FUNCTIONSA function is built upon three fundamental ideas: Domain, Co-domain, and Range.
Domain:
The set of all possible inputs for the function.
Example: For f(x) = x², if x can be any real number, then the domain is all real numbers R.
Co-domain:
The set in which all outputs of the function are defined to lie.
Example: In f(x) = x², co-domain can be chosen as R, since squares are real numbers.
Range:
The set of actual outputs produced by the function from given inputs.
Example: For f(x) = x², the values are always non-negative, so the range = [0,∞).
Thus,
f:Domain → Co-domain, Range ⊆ Co-domain
Example (Mapping):
Domain = {1, 2, 3}
Co-domain = {1, 4, 9, 16}
Range = {1, 4, 9}
Understanding these three terms is crucial because they form the foundation of all types of functions studied in mathematics.

5. Functions can be expressed in different ways:
Algebraic Form: Using equations, e.g., f(x) = 2x+3.
Tabular Form: Showing input-output pairs in a table. Example:



Graphical Form: Plotting points (x,f(x)) on the coordinate plane to study behavior visually.
Conclusion: Different representations provide flexibility in understanding and analyzing
functions. x 1 2 3
f(x) 5 7 9REPRESENTATION OF FUNCTIONS

6. ONE-ONE AND MANY-ONE FUNCTION
One-One (Injective):
Each input maps to a unique output.
Example: f(x)=2x+1.
Diagram: No overlapping arrows.
Many-One:
Two or more inputs map to the same output.
Example: f(x)=x², since f(2) = f(−2) = 4.

7. ONTO AND INTO FUNCTION
Onto (Surjective):
Every co-domain element has a pre-image.
Range = Co-domain.
Example: f(x)=x³ (domain, co-domain = R).
Into:
At least one co-domain element is unmapped.
Range ⊂ Co-domain.
Example: f(x)=x² over R (no negative outputs).

8. POLYNOMIAL FUNCTIONS
Linear: f(x)=ax+b, graph is a straight line.
Quadratic: f(x)=ax² + bx + c, graph is a parabola.
Cubic: f(x)=ax³ + bx² + cx + d, graph is S-shaped.

9. CONCLUSIONFunctions are the backbone of mathematics, as they describe precise input–output relationships. Every function has three core
elements — domain, co-domain, and range — which together determine its nature.
We saw that functions can be represented in algebraic, tabular, or graphical forms, making them versatile for analysis. Further, they
are of different types:
One-One, Many-One
Onto, Into
Polynomial functions (Linear, Quadratic, Cubic)
Special functions like Identity, Constant, and Modulus.
Functions are not just abstract ideas; they have real-life applications in science, economics, engineering, and daily problem-solving.
From predicting population growth to analyzing speed, supply-demand, or profit-loss situations, functions help us model patterns and
relationships accurately.

10. THANK YOU!