#### Let  be a function defined by  where  denotes the greatest integer  Then the range of f is : Option: 1           Option: 2  Option: 3  Option: 4

Piecewise function -

Greatest integer function

The function f: R R defined by f(x) = [x], x R assumes the  value of the greatest integer less than or equal to x. Such a functions called the greatest integer function.

eg;

[1.75] = 1

[2.34] = 2

[-0.9] = -1

[-4.8] = -5

From the definition of [x], we

can see that

[x] = –1 for –1 x < 0

[x] = 0 for 0 x < 1

[x] = 1 for 1 x < 2

[x] = 2 for 2 x < 3 and

so on.

Properties of greatest integer function:

i) [x] ≤ x < [x] + 1

ii) x - 1 < [x] < x

iii) I ≤ x < I+1 ⇒ [x] = I where I belongs to integer.

iv) [[x]]=[x]v)

v)

vi) [x] + [-x] = 2x if x belongs to integer

2[x] + 1 if x doesn’t belongs to integer

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

All possible values of x for f(x) to be defined is known as a domain. If a function is defined from A to B i.e. f: A?B, then all the elements of set A is called Domain of the function.

If a function is defined from A to B i.e. f: A?B, then all the elements of set B are called Co-domain of the function.

The set of all possible values of  f(x) for every x belongs to the domain is known as Range.

For example, let A = {1, 2, 3, 4, 5} and B = {1, 4, 8, 16, 25, 64, 125}. The function f : A -> B is defined by f(x) = x3. So here,

Domain : Set A

Co-Domain : Set B

Range : {1, 8, 27, 64, 125}

The range can be equal to or less than codomain but cannot be greater than that.

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Correct Option (4)