3.8 Greatest and least integer functions  (Page 3/3)

Graph of {x}

Few function expressions in different intervals are :

$$\text{For}-2\le x<-\mathrm{1,}f\left(x\right)=\left\{x\right\}=x-\left[x\right]=x-\left(-2\right)=x+2$$

$$\text{For}-1\le x<\mathrm{0,}f\left(x\right)=\left\{x\right\}=x-\left[x\right]=x-\left(-1\right)=x+1$$

$$\text{For}0\le x<\mathrm{1,}f\left(x\right)=\left\{x\right\}=x-\left[x\right]=x-0=x$$

$$\text{For}1\le x<\mathrm{2,}f\left(x\right)=\left\{x\right\}=x-\left[x\right]=x-1$$

$$\text{For}2\le x<\mathrm{3,}f\left(x\right)=\left\{x\right\}=x-\left[x\right]=x-2$$

The graph of the function is shown here :

We see that there is no restriction on values of x and as such its domain has the interval equal to that of real numbers. The fractional part function can only evaluate to non-negative values between 0≤y<1. Hence,

$$\text{Domain}=R$$

$$\text{Range}=0\le y<1$$

FPF is a periodic function. The values are repeated with a period of 1. Further, function is defined for all real x, but graph is not continous. It breaks at integral values of x.

Least integer function

We have seen that greatest integer function represents the integer, which can be considered to be the floor integral value of a real number. Correspondingly, we define a ceiling function called “least integer function (LIF)”, which returns the least integer greater than or equal to the number (x). We denote least integer function as “[x)” or "(x)". Some authors reserve "(x)" for near integer function. It is not important as we can always specify what we mean by qualifying the symbol explicitly. We interpret LIF as :

• [x) = least integer greater than or equal to the number x
• [x) = least integer not less than or equal to the number x

Clearly, least integer function returns a value, which is the integral “ceiling” of the number. For this reason, least integer function is also known as “ceiling” function. Working rules for finding least integer function are :

• If “x” is an integer, then [x) = x.
• If “x” is not an integer, then [x) evaluates to least integer greater than “x”.

The value of f(x) is an integer (n) such that :

$$f\left(x\right)=n;\text{if}n-1<x\le nn\in Z$$

Graph of least integer function

Few initial values of the functions are :

$$For-3<x\le -\mathrm{2,}f\left(x\right)=\left[x\right)=-2$$

$$For-2<x\le -\mathrm{1,}f\left(x\right)=\left[x\right)=-1$$

$$For-1<x\le \mathrm{0,}f\left(x\right)=\left[x\right)=0$$

$$For0<x\le \mathrm{1,}f\left(x\right)=\left[x\right)=1$$

$$For1<x\le \mathrm{2,}f\left(x\right)=\left[x\right]=2$$

We see that there is no restriction on values of “x” and as such its domain has the interval equal to that of real numbers. On the other hand, the least integer function evaluates only to integer values. It means that the range of the function is set of integers, denoted by "Z". Hence,

$$\text{Domain}=R$$

$$\text{Range}=Z$$

GIF is not a periodic function. Though function is defined for all real x, but graph is not continous. It breaks at integral values of x.

Important properties

Certain properties of least integer function are presented here :

1: If and only if “x”is an integer, then :

$$\left[x\right)=x$$

2: If and only if at least either “x” or “y” is an integer, then :

$$[x+y)=\left[x\right)+\left[y\right)$$

For example, let x = 2.27 and y = 0.63. Then,

$$\Rightarrow [x+y)=[2.27+0.63)=\left[2.9\right)=3$$

$$\Rightarrow \left[x\right)+\left[y\right)=\left[2.27\right)+\left[0.63\right)=3+1=4$$

However, if one of two numbers is integer like x = 2 and y = 0.63, then the proposed identity as above is true.

4: If “x” belongs to integer set, then :

$$\left[x\right)+[-x)=0;x\in Z$$

For example, let x = 2.Then

$$\Rightarrow \left[2\right)+[-2)=2-2=0$$

We can use this identity to test whether “x” is an integer or not?

3: If “x” does not belong to integer set, then :

$$\left[x\right)+[-x)=+1;x\notin Z$$

For example, let x = 2.7.Then

$$\Rightarrow \left[2.7\right)+[-2.7)=3-2=+1$$

Nearest integer function

Nearest integer function, as the name suggests, returns the nearest integer. It is denoted by the symbol, "(x)".

The value of "(x)" is an integer "n" such that :

$$f\left(x\right)=\left(x\right)=n;\text{if}n\le x\le n+1/\mathrm{2,}n\in Z$$

$$f\left(x\right)=n+1;\text{if}n+1/2<x\le n+\mathrm{1,}n\in Z$$

Examples :

$$\left(2.3\right)=\mathrm{2,}\left(2.6\right)=3$$

$$\left(-2.3\right)=-\mathrm{2,}\left(-2.6\right)=-3$$

Exercise