#### Topics

##### Sets and Relations

##### Functions

##### Complex Numbers 33

##### Sequences and Series

##### Locus and Straight Line

##### Determinants

##### Limits

##### Continuity

##### Differentiation

##### Partition Values

##### Measures of Dispersion

##### Skewness

##### Bivariate Frequency Distribution and Chi Square Statistic

##### Correlation

##### Permutations and Combinations

- Introduction of Permutations and Combinations
- Fundamental Principles of Counting
- Concept of Addition Principle
- Concept of Multiplication Principle
- Concept of Factorial Function
- Permutations
- Permutations When All Objects Are Distinct
- Permutations When Repetitions Are Allowed
- Permutations When All Objects Are Not Distinct
- Circular Permutations
- Properties of Permutations
- Combination
- Properties of Combinations

##### Probability

##### Linear Inequations

##### Commercial Mathematics

- Function, Domain, Co-domain, Range
- Types of function

1. One-one or One to one or Injective function

2. Onto or Surjective function - Representation of Function
- Graph of a function
- Value of funcation
- Some Basic Functions - Constant Function, Identity function, Power Functions, Polynomial Function, Radical Function, Rational Function, Exponential Function, Logarithmic Function, Trigonometric function

## Definition

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.

## Notes

In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element.

If f is a function from A to B and (a, b) ∈ f, then f (a) = b, where b is called the image of a under f and a is called the preimage of b under f.

The function f from A to B is denoted by f: A → B.

A funcion is a connection between 2 sets A and B f: A→B such that

1) All elements in A are associated to some element in B

2) This association is unique, that means one and only one.

Let's try to understand this with a simple anology,

Here, let's say `"X"_1` is a set of all children and `"X"_2` is a set of all womens. And `"X"_1` and `"X"_2` have connection as mother and children.

So as per the definition there is a connention between 2 sets `"X"_1` and `"X"_2` such that all the elements of `"X"_1` are associated to some element in set `"X"_2` i.e all the childrens are related to a particualr mother, and this association is unique because no one child can have two or more mothers, but a mother can have more than one child.

Consider the sets D and Y related to each ther as shown, clearly every element in the set D is related to exactly one element in the set Y. So the given relation is a function. f: D → Y.

Here, D is the domain of the function and Y is the co domain of the function.

f(1)= 5

Here, 5 is called the image of 1 under f and 1 is called preimage of 5 under f.

The range of this function is, Range= {2,3,5,7}

The range is a set of real numbers so the function is Real valued function.