GREATEST INTEGER FUNCTION

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

Value of y = [x]

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]  =  1

[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

Graphing Greatest Integer Function

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)

How to plot the above points on the graph ?

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.

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us : 

v4formath@gmail.com

We always appreciate your feedback. 

You can also visit the following web pages on different stuff in math. 

WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations 

Word problems on linear equations 

Word problems on quadratic equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation 

Word problems on unit price

Word problems on unit rate 

Word problems on comparing rates

Converting customary units word problems 

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles 

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems 

Profit and loss word problems 

Markup and markdown word problems 

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed 

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS 

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6