Question

Let $A=Z\setminus \left\{0\right\}$ ie, the set of all non zero integers and $f:A\to R$ (the set of real numbers) be defined by $f\left(x\right)=\frac{\left|x\right|}{x},x\in A$. Find the range and type of the function. Is it one-to-one?

Answer 1

$A=Z-\left(0\right)$
$=\{.....,-3,-2,-1,1,2,3,....\}$, $f\left(x\right)=\frac{\left|x\right|}{x}$
$f\left(1\right)=\frac{\left|1\right|}{1}=\frac{1}{1}=1$
$f\left(2\right)=\frac{\left|2\right|}{2}=\frac{2}{2}=1$
$f(-1)=\frac{1-11}{-1}=\frac{1}{-1}=-1$
$f(-2)=\frac{|-2|}{2}=\frac{-2}{2}=-1$
$f\left(k\right)=\frac{\left|k\right|}{k}=1$ $f(-k)=\Delta$
$\therefore$ the range $=$ {1,-1).
The function is not onto.
The function is not one-to-one function.