Greatest Integer Function

Have you ever sent a package to anyone? If yes, then you would have noticed that the cost of sending your package would be according to the weight of your package. Let us assume that a nearby courier service charges you the following rates that are given below to ship your package to a particular destination.

The shipping cost will be directly proportional to the weight of the package. But, did you notice the weight 1 kg is available in two slabs of pricing? And most of the weights are also available on two slabs? Wondering why it has two options available? Let us learn this with an example. Let us consider the weight of the package to be 2.6 kilos. The cost of shipping this package would be Rs. 150 /-. Now, what would the shipping cost of 2.1 kilos be? It would again be Rs. 150/- itself. Now, this happens due to the greater integer function. In other words, the integer function is also called the step function. There are different ways to represent the greater integer function like these: [ x ], [[ x ]], ⌊ x ⌋.

But What is a Greater Integer Function?

The greater integer function is a function that gives the output of the greatest integer that will be less than the input or lesser than the input. The output is based on the input and there are two rules that need to be followed while writing the output:

• The output is going to be an integer if the input is an integer.

• In case the input isn’t an integer the output is going to be equal to the next smallest integer.

We can consider the second rule as taking the input and writing the closest number or rounding it off to the closest number. To understand the concept better here is an example. Let us consider a number, say 42. Now, with the help of the greater integer function plug in the number to it to find out the result. The output here is going to be 42 since 42 is an integer. Let’s say we have to find the greater integer of a non-integer function. Assume 6.32 to be plugged into the greater integer function. Now, the output will be the next smallest integer. In this case, it is 6.  This example will clear your mind. Here are some more examples to solidify your understanding. 

• ⌊ 32 ⌋ = 32

• ⌊ 14.938 ⌋ = 14 

• ⌊ 9.56 ⌋ = 9

• ⌊ 7 ⌋ = 7

• ⌊ 11.111111 ⌋ = 11

Greater Integer Function Graph

Now that we have understood the concept of greater integer function, let’s focus on how to graph this function. You must be wondering how the graph would look like, right? Well, here’s a hint - the other name for the greater integer function is a step function. Got a hang of it? Let’s understand it better with an example. The graph shown below is how your greater integer function looks like. The dark point on the left of a step represents that it is a member of the graph and the light point indicates that the point isn’t a part of a graph. Note that between every interval the function f(x) remains the same and the value of the function remains constant within an interval. 

Solved Problems 

Question 1: Find the greatest integer function for:

( i ) ⌊ -2 ⌋ 

( ii ) ⌊ 1.32 ⌋ 

( iii ) ⌊ -14.982 ⌋ 

( iv ) ⌊ 3.98792 ⌋ 

( v ) ⌊ -8.208322 ⌋ 

( vi ) ⌊ 140 ⌋ 

Solution: 

( i ) Let us consider ⌊ -2 ⌋ on a number line. This number lies between - 1 and - 3.  When plugged into the largest integer function, the result is going to be -2. 

( ii ) Let us consider ⌊ 1.32 ⌋ on a number line. This number lies between  1 and 2.  When plugged into the largest integer function, the result is going to be 1. 

( iii ) Let us consider ⌊ -14.982 ⌋ on a number line. This number lies between - 14 and - 15.  When plugged into the largest integer function, the result is going to be -15. 

( iv ) Let us consider ⌊ 3.98792 ⌋ on a number line. This number lies between 3 and 4.  When plugged into the largest integer function, the result is going to be -2. 

( v ) Let us consider ⌊ 8.208322 ⌋ on a number line. This number lies between - 9 and - 8.  When plugged into the largest integer function, the result is going to be 8. 

( vi ) Let us consider ⌊ 140 ⌋ on a number line. This number lies between 139 and 141.  When plugged into the largest integer function, the result is going to be 140.