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If $h\left( x \right) = \left[ {\ln \frac{x}{e}} \right] + \left[ {\ln \frac{e}{x}} \right]$ ,where [.] denotes greatest integer function, then which of the following is false ?
Range of $h(x)$ is $\{-1, 0\}$
$h(x)$ is a periodic function
If $h(x) = -1$ , then $x$ can be rational as well as irrational
If $h(x) = 0$ , then $x$ must be irrational
Solution
$h(x)=\left[\ln \frac{x}{e}\right]+\left[\ln \frac{e}{x}\right]=[t]+[-t], t=\ln \left(\frac{x}{e}\right)$
$=\left\{\begin{array}{cc}{0} & {\ln \frac{x}{e} \in I} \\ {-1} & {\ln \frac{x}{e} \notin I}\end{array}\right.$
Range is $\{-1,0\}$
If $\mathrm{h}(\mathrm{x})=0 \Rightarrow \frac{\mathrm{x}}{\mathrm{e}} \in \mathrm{k}, \mathrm{k} \in \mathrm{I}$
$\Rightarrow x$ can be rational or irrational
Similar Questions
Let $\quad E_1=\left\{x \in R : x \neq 1\right.$ and $\left.\frac{x}{x-1}>0\right\}$ and $\quad E_2=\left\{x \in E_1: \sin ^{-1}\left(\log _e\left(\frac{x}{x-1}\right)\right)\right.$ is a real number $\}$.
(Here, the inverse trigonometric function $\sin ^{-1} x$ assumes values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ )
Let $f : E _1 \rightarrow R$ be the function defined by $f(x)=\log _c\left(\frac{x}{x-1}\right)$ and $g: E_2 \rightarrow R$ be the function defined by $g(x)=\sin ^{-1}\left(\log _e\left(\frac{x}{x-1}\right)\right)$
| $LIST I$ | $LIST II$ |
| $P$ The range of $f$ is | $1$ $\left(-\infty, \frac{1}{1- e }\right] \cup\left[\frac{ e }{ e -1}, \infty\right)$ |
| $Q$ The range of $g$ contains | $2$ $(0,1)$ |
| $R$ The domain of $f$ contains | $3$ $\left[-\frac{1}{2}, \frac{1}{2}\right]$ |
| $S$ The domain of $g$ is | $4$ $(-\infty, 0) \cup(0, \infty)$ |
| $5$ $\left(-\infty, \frac{ e }{ e -1}\right]$ | |
| $6$ $(-\infty, 0) \cup\left(\frac{1}{2}, \frac{ e }{ e -1}\right]$ |
The correct option is:
