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1.Relation and Function

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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 ?

A

Range of $h(x)$ is $\{-1, 0\}$

B

$h(x)$ is a periodic function

C

If $h(x) = -1$ , then $x$ can be rational as well as irrational

D

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

Std 12

Mathematics

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