Piecewise function - Practice Questions & MCQ

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