# Functions or Mappings

Let A and B be two non-empty sets. Then, a rule or a correspondence *f *which associates to each Â , a unique element Â is called a function or a mapping from A to B and we write, *Â f: A **Ã ** B. f (x) *is called the image of x and x is called the pre-image of *f (x).*

A is called the domain of *f *and Â is called the range of *f.*

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# Various types of functions

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* I. *A functionÂ

*f : A*

*Ã*

*B*is said to be:

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1.Â **Manyâ€“one â€“ **If two or more than two elements in A have the same image in B.

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2.Â **Oneâ€“One â€“ **If distinct elements in A have distinct images in B i.e.

*f*is one-one, if

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3.Â **Into â€“** If at least one element in B which has no pre-image in A.

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One-one mapping is calledÂ *injective;Â *onto mapping is calledÂ *surjectiveÂ *and a one-one, onto mapping is calledÂ *bijective.Â *

4.Â **Onto or Surjective functions â€“ **Let f : A Ã B. If every element in B has at least one preâ€“image in A, then

*f*is said to be an onto function.

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5.Â **Identity function â€“*** *Let A be a nonâ€“empty set. Then, the function I defined by I : A Ã A : I (x) = x for all Â is called an identity function on A. Â It is a oneâ€“toâ€“one onto function with domain A and range A.

6.Â **Constant function â€“ **Let f : A Ã B, defined in such a way that all the elements in A have the same image in B, then

*f*is said to be a constant function.

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7.Â **Inverse function â€“**

*Let f be a oneâ€“one onto function from A to B. Let y be an arbitrary element of B. Then*

*f*being onto, there exists an element x A such that f (x) = y.

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A function is invertible if and only if f is oneâ€“one onto.

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**
II .Â Â Â (i) Even functions â€“ **A function

*f (x)*is said to be even if

*f (-x) = f (x).*

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*Â Â Â Â ***(ii) Odd functions*** â€“ *A function

*f (x)*is said to be odd if

*f (-x) = -f (x).*

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**III.Â Periodic functions â€“ **A function

*f (x)*is said to be periodic with period

*p,*if

*f (x)=f(x+p)*

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