Question

If $f\left(x\right)={\log }_{(x-1)}\frac{|x|}{x}$, where $[.]$ denotes greatest integer function, then

• A. domain of $f=(2,\infty )$
• B. range of $f=\{0,1\}$
• C. domain of $f=[3,\infty )$
• D. range of $f=\left\{0\right\}$

Answer 1

Answer: (A), (D)

The correct options are
A domain of $f=(2,\infty )$
D range of $f=\left\{0\right\}$

Given,

$f\left(x\right)={\log }_{x-1}\left(\frac{\left|x\right|}{x}\right)$

${\log }_{f\left(x\right)}g\left(x\right)\Rightarrow g\left(x\right)>0$ and $(0<f(x)<1)$ or $f\left(x\right)>1$

Domain $f\left(x\right)=(1,2)\cup (2,\infty )$ as $1<x<2,x>2$

For $1<x<2\to f\left(x\right)=0$ and

for $x>1\to f\left(x\right)=0$

Range $f\left(x\right)=0$