Question

Find the domain and range of the following functions.

(i) $\quad f(x)=\cos ^{-1} \sqrt{\log _{|x|} \frac{|x|}{x}}$, where [.] denotes the greatest integer function.

(ii) $f(x)=\sqrt{\log _{1 / 2} \log _2\left[x^2+4 x+5\right]}$ where $[.1$ denotes the greatest integer function

(iii) $f(x)=\sin ^{-1}\left[\log _2\left(\frac{x^2}{2}\right)\right]$, where [.] denotes greatest integer function.

(iv) $f(x)=\log _{[x-1]} \sin x$, where [] denotes greatest integer function .

(v) $f(x)=\tan ^{-1}(\sqrt{[x]+[-x]})+\sqrt{2-|x|}+\frac{1}{x^2}$, (where [] denotes greatest integer function)

Answer 1

SULOTION : (i) Fordomain (i) $[x]>0$ and $[x] \neq 1$ so $[x] \geq 2$, so $x \in[2, \infty)$

for range if $x \in[2, \infty)$, then $\frac{|x|}{x}=1 \quad$ so $f(x)=\cos ^{-1} 0=\frac{\pi}{2}$

Range of $f(x)=\left\{\frac{\pi}{2}\right\}$

(ii)

$f(x)=\sqrt{\log _{1 / 2} \log _2\left[x^2+4 x+5\right]} \\$

$\text { D : } 0<\log _2\left[x^2+4 x+5\right] \leq 1 \\$

$\text { or } \quad 1<\left[x^2+4 x+5\right] \leq 2 \Rightarrow\left[x^2+4 x+5\right]=2 \\$

$\text { or } \quad 2 \leq x^2+4 x+5<3$

or $1<\left[x^2+4 x+5\right] \leq 2 \quad \Rightarrow \quad\left[x^2+4 x+5\right]=2$

or $2 \leq \mathrm{x}^2+4 \mathrm{x}+5<3$

$D: x \in(-2-\sqrt{2},-3] \cup[-1,-2+\sqrt{2})$

$R:\{0\}$

(iii)$f(x)=\sin ^{-1}\left[\log _2\left(\frac{x^2}{2}\right)\right] \Rightarrow-1 \leq \log _2\left(\frac{x^2}{2}\right)<2 \Rightarrow \frac{1}{2} \leq \frac{x^2}{2}<4 \\$

$\Rightarrow \quad x \in(-\sqrt{8},-1] \cup[1, \sqrt{8}) \quad \text { and } R:\left\{-\frac{\pi}{2}, 0, \frac{\pi}{2}\right\}$

(iv)$f(x)=\log _{[x-1]} \sin x \\$

$\sin x>0 \Rightarrow x \in(2 n \pi,(2 n+1) \pi) \\$

$\text { here } \quad[x-1]>0 \&[x-1] \neq 1 \quad \Rightarrow \quad x \in[3, \infty]$

Domain $x \in[3, \pi) \cup\left[\bigcup_{n-1}^n(2 n \pi,(2 n+1) \pi)\right]$.

For range $\sin x \in(0,1]$ and $[x-1] \in[2, \infty)$ so range $\in(-\infty, 0]$

(v) $\quad f(x)=\tan ^{-1} \sqrt{[x]+[-x]}+\sqrt{2-|x|}+\frac{1}{x^2}$

Domain: (i) $\quad[x]+[-x] \geq 0 \quad \Rightarrow \quad x \in I$

(ii) $2-|x| \geq 0 \Rightarrow|x| \leq 2 \Rightarrow x \in[-2,2]$

(iii) $x \neq 0$

For domain (i) $\cap($ ii) $\cap$ (iii)

Domain : $\{-2,-1,1,2\}$

Range : $\left\{\frac{1}{4}, 2\right\}$