this is the ppt based on types of function and types of function

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1. HELLO EVERYONE!
Thisprojectisto giveinformationaboutFunctions
and TypesofFunctions.

2. FUNCTIONS AND TYPES OF FUNCTIONS.
By
SAGAR KUMAR SONI

3. WHAT IS A FUNCTION?

A functionisa specialrelationship betweentwo sets where:each input(from the
firstset,called domain) hasexactly oneoutput (from thesecond 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 FUNCTIONS

Afunctionisbuiltuponthree fundamental ideas: Domain,Co-domain,andRange.
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. Graphical Form:Plottingpoints(x,f(x))onthecoordinateplanetostudybehavior visually.
Conclusion: Differentrepresentationsprovideflexibilityinunderstandingand analyzing
functions.
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:
x 1

2 3

f ( x ) 5 7 9
REPRESENTATION 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. CONCLUSION
Functions are the backbone of mathematics,asthey describe preciseinput–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!