Question

The domain of definition of the function, $f(x)=[2 \tan \pi x]^{\log [2 \tan \pi x]}\left(\frac{x^2+2 x-3}{4 x^2-4 x-3}\right)$ where [ ] denotes the greatest integer function is $\left[ n +\frac{1}{4}, n +\frac{1}{2}\right) ; n \in I$ then

• A. $n =0$
• B. $n \leq-4$
• C. $n \geq 4$
• D. none

Answer 1

Answer: (A), (B), (C)

$f ( x )= a ^{\log _{ a } N} N$ and $0<[2 \tan \pi x ]$ or $[2 \tan \pi x ]>1$
$\text { and } \frac{x^2+2 x-3}{4 x^2-4 x-3}>0 \text { i.e. } \frac{(x+3)(x-1)}{(2 x-3)(2 x+1)}>0 $
$\Rightarrow x \in(-\infty,-3) \cup(-1 / 2,1) \cup(3 / 2, \infty) $.....(i)
$\text { now } 0< [2 \tan \pi x]< 1 \text { not possible } \therefore[2 \tan \pi x] >1$ $\Rightarrow 2 \tan \pi x \geq 2 \Rightarrow \tan \pi x \geq 1$
$\Rightarrow n \pi+\frac{\pi}{4} \leq \pi x< n \pi+\frac{\pi}{2} n \in I$
$\Rightarrow n+\frac{1}{4} \leq x< n+\frac{1}{2}$<....(ii)br/> Common solution of (i) and (ii) possible only if $n=0, n \geq 2$ or $n \leq-4$ ]