Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Diagrammatic representation of function

If f: X \rightarrow Y can be written as

f=\ {(x, f(x)) : x \in X\}

where  x is called variable.

f^ {-1} (B)= \{x \in X : f(x) \in B\}

Example:

Let  A = \{1,2,3,4,5\}, B =\{a,b,c\}

Consider  f_1 = \{(1,a), (2,b), (3,b),(4,a),(5,b)\}

 f_1 \subset A \times B

Hence  \text {dom f}=\{1,2,3,4,5,\} =A

Codomain of f=\{a,b,c\}= B

\text {rng } f_1 = \{a,b\} \subset B

Moreover every element of   A has a unique image in  B . Hence f_1 is a function from  A to B .

Now for the above sets  A and  B , consider:

 f_2= \{(1,b),(2,a),(3,c),(5,a)\}

and

f_3= \{(1,a),(2,b),(3,a),(4,b),(5,c),(1,c)\}

Hence f_2 is not a function from  A to  B since  \text {dom } f_1= \{1,2,3,5\}\neq A has no image in  B .

Also f_3 is not a function from  A to  B since the element  1 \in A has two different images i.e.,  a and  c, i.e.,two different ordered pairs  (1,a) and (1,c) in the  f have the same first component.

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