Chapter SH, Problem 3.20EP

To determine

To graph the function $f(x)=2[[x]]$ .

Answer to Problem 3.20EP

Graph of f(x)=2[[x]]

  

Domain is the complete set of real numbers i.e. R .

Range is the set of integers Z .

Explanation of Solution

Given information:

The function f(x)=2[[x]].

Calculation:

Function f(x)=2[[x]].

The greatest integer function f(x)=2[[x]] is defined as ; $[[x]]$ = integer less than or equal to x

Substituting x=0 in the equation

  $f(x)=2[[0]].$

Greatest integer less than or equal to 0 is 0.

So,

  $f(x)=2(0)$

On solving,

  $f(x)=0$

Substituting $x=1$in the equation

  $f(x)=2[[1]].$

Greatest integer less than or equal to 1 is 1.

So,

  $f(x)=2(1)$

On solving,

  $f(x)=2$

Substituting $x=2$ in the equation

  $f(x)=2[[2]]$ .

Greatest integer less than or equal to 2 is 2.

So,

  $f(x)=2(2)$

On solving,

  $f(x)=4$

Substituting $x=-1$in the equation

  $f(x)=2[[-1]].$

Greatest integer less than or equal to -1 is -1.

So,

  $f(x)=2(-1)$

On solving,

  $f(x)=-2$

Putting values of x and y in the table form

  

The graph of the function $f(x)=2[[x]]$is given below

  

Domain D(f)

The function $f(x)=2[[x]]$is defined for all real numbers, therefore its domain D(f) is the complete set of real numbers i.e. R

Range

The function $f(x)=2[[x]]$ is an integer - valued function but many - one, onto function, because every positive integer m can be obtained in several ways as : $[m]=m=[m.r{r}^{\text{'}}{}^{}{r}^{\text{'}\text{'}}\mathrm{......}]$where r’s after decimal, are all any integers. Similarly for negative integers. Therefore range of the function $f(x)=[x]$ is the set of integers Z .