The greatest integer function is denoted by

**y = [x]**

For all real values of '*x'*, the greatest integer function returns the greatest integer which is less than or equal to 'x'.

In essence, it rounds down to the the nearest integer.

That is,

[3] = 3

[3.2] = 3

[3.9] = 3

[-2] = -2

[-2.8] = -3

In the greatest integer function, [x], if 'x' is integer, what should we do ?

In [x], if 'x' is integer, then the value of [x] is also the same value of 'x'.

**Examples : **

[1] = 1

[4] = 4

[-2] = -2

[-7] = -7

[128] = 128

[-225] = -225

In the greatest integer function, [x], if 'x' is non integer (decimal), What should we do ?

In [x], if 'x' is non integer(decimal), then the value of [x] is the largest integer less than or equal to the given decimal.

To understand of this, let us do this with number line.

**Example 1 :**

[1.6] = ?

To have value of [1.6], we have to mark '1.6' on the number line as shown below.

Pick the nearest integer on the left side of '1.6'.

That is '1'.

So,

[1.6] = 1

**Example 2 :**

[4.5] = ?

To have value of [4.5], we have to mark '4.5' on the number line as shown below.

Pick the nearest integer on the left side of '4.5'.

That is '4'.

So,

[4.5] = 4

**Example 3 :**

[5.6] = ?

To have value of [5.6], we have to mark '5.6' on the number line as shown below.

Pick the nearest integer on the left side of '5.6'.

That is '5'.

So,

[5.6] = 5

**Example 4 :**

[-2.3] = ?

To have value of [-2.3], we have to mark '-2.3' on the number line as show below below.

Pick the nearest integer on the left side of '-2.3'.

That is '-3'.

So,

[-2.3] = -3

**Example 5 :**

[-5.6] = ?

To have value of [-5.6], we have to mark '-5.6' on the number line as shown below.

Pick the nearest integer on the left side of '-5.6'.

That is '-6'.

**So, **

[-5.6] = -6

**Example 6 :**

[-7.8] = ?

To have value of [-7.8], we have to mark '-7.8' on the number line as shown below.

Pick the nearest integer on the left side of '-7.8'.

That is '-8'.

So,

[-7.8] = -8

To graph the greatest integer function

y = [x],

we have to substitute some random values for 'x'.

Let x = -3, -2, -1, 0, 1, 2, 3.

Then,

y = [-3] = -3 -----> (-3, -3)

y = [-2] = -2 -----> (-2, -2)

y = [-1] = -1 -----> (-1, -1)

y = [0] = 0 -----> (0, 0)

y = [1] = 1 -----> (1, 1)

y = [2] = 2 -----> (2, 2)

y = [3] = 3 -----> (3, 3)

Let us take the point (-3, -3).

Mark the point on xy- plane with a filled circle at (-3, -3).

Then extend a line for '1' unit on the left side of (-3,-3) and end up with empty circle.

Do the same thing for the other points too.

Now you will have a graph as shown below.

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