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

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
**