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Functions

Definition: If each element of a set A is associated with exactly one element of a set B, then this association is called a function from A to B.

       The set A is called the Domain and the set B is called the Co-domain of the function.

                             

Domain = {1, 2, 3, 4, 5}
Co-domain = {a, b, c, d, e, f}
Range = {a, b, c, d, f}


⇒  Total no. of functions from A to B is
       n × n × ..... m times = nm
⇒ Each element of A must be associated.
⇒ All the elements of the set B need not have the association.
⇒ The set of elements of B which are associated with elements of set A is called the Range.
⇒ Range is a subset of the Co-domain 

Different types of functions
One-one function: A function f : A B is called a one-one (or) Injective (or) Monomorphic function if different elements of A have different images in B.
                               

Note: If f is not one-one then it is called a Many-one function.

                  
Onto-function: A function f : A B is called an onto-function if every element of B is an image of some element of A i.e. if Co-domain = Range

Note: If f is not surjective (onto) then it is called an into function.

Bijective function: A function f : A B is said to be bijective (or) one-one onto function from A onto B. If f : A B is both one-one function and onto function.

                                
No. of bijective mappings: If n(A) = m = n(B), then the number of possible bijections from A to B is m!

 

Constant function: A function f : A B is called a constant function, if the range of f consists of only one element i.e. , f(x) = k where k is the fixed element of B.


                                       
Note: 1. A constant function is a surjection if its Co domain is a singleton set.
           2. A constant function is an injection if its domain is a singleton set.
Identity function: The mapping f : A A defined by f(x) = x, i.e. every element of A is f-image of itself is called Identity function on A. It is denoted by IA (or) I.

Inverse function: If f : A B is a one-one onto mapping then f-1-1 : B A is called inverse of the mapping f from A to B.
Note:

        

Composite function: If f : A B and g : B C are two functions then the function gof : A C defined by (gof)(a) = g[f(a)] , is called the composite (or) product function of f and g.
               


Odd function: A function f(x) is said to be an add function if f(-x) = - f(x) ∀ x.

Even function: A function f(x) is said to be an even function if f(-x) = f(x)∀ x


Exponential function: If a > 0, a ≠ 1, a ∈ R then the function f(x) = ax is called an exponential function.
     


Logarithmic function: If a > 0, a ≠ 1, a ∈ R then the function f(x) = logax is called a logarithmic function.

Signum function: It is denoted by y = Sgn(x). It is defined by

               


The domains and ranges of some standard functions


CONCEPTUAL THEOREMS
Theorem 1: If f : A B and g : B C are two one-one (injective) functions then show that the mapping gof : A C is one-one.
Proof: Given f : A B and g : B C are two one-one functions. We have to prove that
gof : A C is also one-one.

    f(a1) = f(a2) [...   g is one-one]
             a1 = a2 [...  f is one-one]
     Hence gof: A C is a one-one function.


Theorem 2: If f : A B and g : B C are two onto (Surjective) functions then show that the mapping gof : A C is onto.
Proof: Given f : A B and g : B C are two onto functions.
            We have to prove that gof : A C is also onto.
            
Let a ∈ A, b ∈ B, c ∈ C
Since g : B C is an onto function.
Therefore, there exists an element b ∈ B such that g(b) = c.
Since f : A B is an onto function.
Therefore, there exists an element a ∈ A such that f(a) = b
Now g(b) = c

 g[f(a)] = c
 (gof)(a) = c
Thus for any element c ∈ C, there is an element a ∈ A such that (gof)(a) = c
Hence, gof : A C is an onto.


                          

Posted Date : 06-11-2020

గమనిక : ప్రతిభ.ఈనాడు.నెట్‌లో కనిపించే వ్యాపార ప్రకటనలు వివిధ దేశాల్లోని వ్యాపారులు, సంస్థల నుంచి వస్తాయి. మరి కొన్ని ప్రకటనలు పాఠకుల అభిరుచి మేరకు కృత్రిమ మేధస్సు సాంకేతికత సాయంతో ప్రదర్శితమవుతుంటాయి. ఆ ప్రకటనల్లోని ఉత్పత్తులను లేదా సేవలను పాఠకులు స్వయంగా విచారించుకొని, జాగ్రత్తగా పరిశీలించి కొనుక్కోవాలి లేదా వినియోగించుకోవాలి. వాటి నాణ్యత లేదా లోపాలతో ఈనాడు యాజమాన్యానికి ఎలాంటి సంబంధం లేదు. ఈ విషయంలో ఉత్తర ప్రత్యుత్తరాలకు, ఈ-మెయిల్స్ కి, ఇంకా ఇతర రూపాల్లో సమాచార మార్పిడికి తావు లేదు. ఫిర్యాదులు స్వీకరించడం కుదరదు. పాఠకులు గమనించి, సహకరించాలని మనవి.

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