f: R R

f(x) = [x]

[x] is the greatest integer less than or equal to x

 

[0] = 0

[0.0001] = 0

[0.1] = 0

[0.9999] = 0

[1] = 1

 

[1.01] = 1

[1.2] = 1

[1.99] = 1

[1.9999999] = 1

[2] = 2

 

[2.0001] = 2

[2.2] = 2

[2.999] = 2

[3] = 3


For negative numbers

 

[–0.1]

Since it is greatest integer less than or equal to x

Integers less than – 0.1 = –1, –2, –3

Greatest Integer less than – 0.1 = –1

∴ [–0.1] = –1

 

[– 0.5]

Since it is greatest integer less than or equal to x

Integers less than – 0.5 = –1, –2, –3

Greatest Integer less than – 0.5 = –1

∴ [–0.5] = –1

 

[–0.999]

Since it is greatest integer less than or equal to x

Integers less than – 0.999 = –1, –2, –3

Greatest Integer less than – 0.999 = –1

∴ [–0.999] = –1

 

[–1]

[–1] = –1

 

[–1.1]

Since it is greatest integer less than or equal to x

Integers less than – 1.1 = –2, –3, –4

Greatest Integer less than –1.1 = –2

∴ [–1.1] = –2

 

[–1.999]

Since it is greatest integer less than or equal to x

Integers less than –1.999 = –2, –3, –4

Greatest Integer less than –1.999 = –2

∴ [–1.999] = –2

 

[–2]

[–2] = –2

 


Now, let us draw the graph

[0] = 0, [0.0001] = 0, [0.1] = 0, [0.9999] = 0, [1] = 1

So, for 0 ≤ x < 1, f(x) = 0

 

[1] = 1, [1.01] = 1, [1.2] = 1, [1.99] = 1, [1.9999999] = 1, [2] = 2

So, for 1 ≤ x < 2, f(x) = 1

 

[2] = 2, [2.0001] = 2, [2.2] = 2, [2.999] = 2, [3] = 3

So, for 2 ≤ x < 3, f(x) = 2

 

[–0.1] = –1, [–0.5] = –1, [–0.999] = –1, [–1] = –1

So, for –1 ≤ x < 0, f(x) = –1

 

[–1.1] = –1, [–1.999] = –2, [–2] = –2

So, for –2 ≤ x < –1, f(x) = –2


To summarise,

For 0 ≤ x < 1, f(x) = 0

For 1 ≤ x < 2, f(x) = 1

For 2 ≤ x < 3, f(x) = 2

For –1 ≤ x < 0, f(x) = –1

For –2 ≤ x < –1, f(x) = –2

Greatest Integer Function.jpg

 

Here,

Domain = All values of x = R

 

Range = All values of y

Since y will have integer values ( … –3, –2, –1, 0, 1, 2, 3, …)

Range = All Integers = Z

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.