-
Engineering and Architecture
Exams
Colleges
Predictors
Resources
-
Computer Application and IT
Quick Links
Colleges
-
Pharmacy
Colleges
Resources
-
Hospitality and Tourism
Colleges
Resources
Diploma Colleges
-
Competition
Other Exams
Resources
-
School
Exams
Top Schools
Products & Resources
-
Study Abroad
Top Countries
Resources
-
Arts, Commerce & Sciences
Colleges
Upcoming Events
Resources
-
Management and Business Administration
Exams
Colleges & Courses
Predictors
-
Learn
Law Preparation
MBA Preparation
Engineering Preparation
Medical Preparation
-
Online Courses and Certifications
Top Streams
Specializations
- Digital Marketing Certification Courses
- Cyber Security Certification Courses
- Artificial Intelligence Certification Courses
- Business Analytics Certification Courses
- Data Science Certification Courses
- Cloud Computing Certification Courses
- Machine Learning Certification Courses
- View All Certification Courses
Resources
-
Medicine and Allied Sciences
Colleges
Predictors
Resources
-
Law
Resources
Colleges
-
Animation and Design
Exams
Predictors & Articles
Colleges
Resources
-
Media, Mass Communication and Journalism
Colleges
Resources
-
Finance & Accounts
Top Courses & Careers
Colleges
Piecewise function - (Concept)
Signum function:
The function f : RR is defined by
$\operatorname{sgn}(\mathrm{x})=\left\{\begin{array}{ccc}1 & \text { if } & x>0 \\ -1 & \text { if } & x<0 \\ 0 & \text { if } & x=0\end{array}\right.$
is called the signum function. The domain of the signum function is R and the range is the set {-1,0,1}.
This function can also be written in another form:
$\operatorname{sgn}(x)=\left\{\begin{array}{c}\frac{|x|}{x}, x \neq 0 \\ 0, x=0\end{array}\right\}$
Graph:
Range $\in\{-1,0,1\}$
Greatest integer function (G.I.F.)
The function $f: R \rightarrow R$ defined by $f(x)=[x], x \in R$ assumes the value of the greatest integer which is equal to or less than $x$. Such a function is called the greatest integer function.
$
\begin{aligned}
& \mathrm{eg} ; \\
& {[1.75]=1} \\
& {[2.34]=2} \\
& {[-0.9]=-1} \\
& {[-4.8]=-5} \\
& {[4]=4} \\
& {[-1]=-1}
\end{aligned}
$
Graph:
From the definition of $[x]$, we can see that
$
\begin{aligned}
& {[x]=-1 \text { for }-1 \leq x<0} \\
& {[x]=0 \text { for } 0 \leq x<1} \\
& {[x]=1 \text { for } 1 \leq x<2} \\
& {[x]=2 \text { for } 2 \leq x<3 \text { and so on. }}
\end{aligned}
$
Properties of greatest integer function:
i) [ a ] = a (If a is an integer)
ii) $[[x]]=[x]$
iii) $x-1<[x] \leq x$
iv) $[x+a]=[x]+a \quad$ (If $a$ is an integer)
v) $[x-a]=[x]-a \quad$ (If $a$ is an integer)
vi) $[x]+[-x]=\left\{\begin{array}{ccc}0, & \text { if } & x \in Z \\ -1, & \text { if } x \notin Z & \end{array}\right.$
Fractional part function:
$
\{x\}=x-[x]
$
When [ x ] is the Greatest Integer Function
$
\begin{aligned}
& \mathrm{Eg} \\
& \{2.2\}=2.2-[2.2]=2.2-2=0.2 \\
& \{1.7\}=1.7-[1.7]=1.7-1=0.7 \\
& \{2\}=2-[2]=2-2=0 \\
& \{-2.2\}=-2.2-[-2.2]=2.2-(-3)=0.8 \\
& \{-1.7\}=-1.7-[-1.7]=1.7-(-2)=0.3 \\
& \{-2\}=-2-[-2]=2-(-2)=0
\end{aligned}
$
Clearly, $0 \leq\{x\}<1 \mid$
Graph
Domain: R
Range $\in[0,1)$
Properties of the fractional part of x
i) $\{x\}=x$ if $0 \leq x<1$
ii) $\{\mathrm{a}\}=0$, if a is an integer
iii) $0 \leq\{x\}<1$
iv) $\{x+a\}=\{x\} \quad$ (If $a$ is an integer)
v) $\{x\}+\{-x\}=1$, if $x$ doesn't belongs to integer
vi) $\{x\}+\{-x\}=0$, if $x$ belongs to integer
| Exam | Chapter |
| JEE MAIN | Sets, Relations and Functions |
If $f(x)=\frac{|x|}{x}$ and $g(x)=\operatorname{sgn}(x)$. Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $f(x)=3 \operatorname{sgn}(x)+1 / 2 \operatorname{sgn}(x)$.. Then range of $\mathrm{f}(\mathrm{x})$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is the range of f(x) = 4[x]?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the details of the function [x] +[-x]
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $f(x)= [x]/2$; what is the last positive value of $|f(x)|$?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If f(x) = [x] -x . Then the range of f(x) is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is the range of$ f(x) = 5 {x} $?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is the range of function f(x) = {x}+{-x}
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f:(1,3) \rightarrow R_{\text {be a function defined by }} f(x)=\frac{x[x]}{1+x^2}$, where $^{[x]}$ denotes the greatest integer $\leq x$. Then the range of f is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $[t]$ denote the greatest integer $\leq t$ and $\lim _{x \rightarrow 0} x\left[\frac{4}{x}\right]=A$. Then the function, $f(x)=\left[x^2\right] \sin (\pi x)$ is discontinuous, when x is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let [t] denote the greatest integer. Then the equation in x, has.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The real valued function where
denotes the greatest integer less than or equal to x, is defined for all x belonging to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If . Then what is the range of f(x)?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For , let
denote the greatest integer
then the sum of the series
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let [x] denote the greatest integer where
. If the domain of the real-valued function
is.
, then the value of a+b+c is?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let denote the greatest integer less than or equal to
. Then, the values of
satisfying the equation
lie in the interval:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If be the greatest integer less than or equal to
then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and the equation
(where
denotes the greatest integer
) has no integral solution, then all possible values of a lie in the interval :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For , let
denote the greatest integer
, then the value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If . Find the value of
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$[3.6]-[-2.2]+[5]=$ ? where $[$. $]$ stands for the greatest integer function.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\operatorname{Sgn}(5)=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which function represents the following graph?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If$[x]^2 - 5[x] + 6 = 0$, where [. ] denote the greatest integer function, then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Solve $\operatorname{sgn}\left(\frac{x-1}{x}\right)>-1 .$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The graph $y=\left \{ 2x \right \}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of solutions of $[x]+x=2\{x\}$ are:
(where [. ] is the greatest integer function, and $\{$.$\}$ is fraction part of $x$ )
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the number of real values of $x$ satisfying
$
[x]\left(\frac{[x]}{[x-2]}-\frac{[x-2]}{[x]}\right)=\frac{4\{x\}+8}{[x-2]},
$
where $[x]$ denotes the greatest integer (floor) function and $\{x\}$ denotes the fractional part of $x$.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The solution of $2+x=2[x]+3\left \{ x \right \}$ is
(where [ . ] is the greatest integer function, and { . } is fractional part of x)
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Range of $\{x\}, x \in[-1,5]$, where {$.$\} stands for fractional part function, is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\mathbb{R}$ be the set of real numbers and $f: \mathbb{R} \rightarrow \mathbb{R}_{\text {be defined }}$ $f(x)=\frac{\{x\}}{1+[x]^2}$, where $[x]_{\text {is the greatest integer less than or equal to } x \text { and }\{x\}=x-[x]}$ Which of the following statements are true?
1. The range of $f$ is a closed interval.
II. $f$ is continuous on $\mathbb{R}$
III. $f$ is one-one on $\mathbb{R}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $A$ denote the set of all real numbers $x$ such that $x^3-[x]^3=(x-[x])^3$, where $[x]$ is the greatest integer less than or equal to $x$. Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Signum function The function
f:RR defined by
