Question

Find the domain of definitions of the following functions :

(i) $\quad f(x)=\sqrt{3-2^x-2^{1-x}}$

(ii) $\quad f(x)=\sqrt{1-\sqrt{1-x^2}}$

(iii) $\quad f(x)=\left(x^2+x+1\right)^{-3 / 2}$

(iv) $\quad f(x)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}}$

(v) $\quad f(x)=\sqrt{\tan x-\tan ^2 x}$

(vi) $\quad f(x)=\frac{1}{\sqrt{1-\cos x}}$

(vii) $f(x)=\sqrt{\log _{1 / 4}\left(\frac{5 x-x^2}{4}\right)}$

(vii) $\quad f(x)=\log _{10}\left(1-\log _{10}\left(x^2-5 x+16\right)\right)$

Answer 1

SOLUTION : (i) $f(x)=\sqrt{3-2^x-2 \cdot 2^{-x}} \\$ 

$3-2^x-2 \cdot 2^{-x} \geq 0 \\$

$\text { or } \left(2^x\right)^2-3 \cdot 2^x+2 \leq 0 \\$

$\text { or }\left(2^x-1\right)\left(2^x-2\right) \leq 0 \quad \Rightarrow \quad 2^x \in[1,2] \quad \Rightarrow \quad x \in[0,1]$

$\left(2^x\right)^2-3 \cdot 2^x+2 \leq 0$

(ii) $f(x)=\sqrt{1-\sqrt{1-x^2}}$ 

$\Rightarrow \quad 2^x \in[1,2] \quad \Rightarrow \quad x \in[0,1]$

$1-\sqrt{1-x^2} \geq 0 \Rightarrow  \sqrt{1-x^2} \leq 1 \\$

$f(x)=\left(x^2+x+1\right)^{-3 / 2} \Rightarrow  D: x \in R$

$\quad \Rightarrow \quad 0 \leq 1-x^2 \leq 1 \quad \Rightarrow \quad x \in[-1,1]$

(iii) $f(x)=\left(x^2+x+1\right)^{-3 / 2} \Rightarrow D: x \in R$

(iv) $f(x)=\sqrt{\frac{x-2}{x+2}}+\sqrt{\frac{1-x}{1+x}} \Rightarrow  \frac{x-2}{x+2} \geq 0 \text { and } \frac{1-x}{1+x} \geq 0 \\$

$x \in(-\infty,-2) \cup[2, \infty) \text { and }  x \in(-1,1] \\$

$D: \phi $

(v) $f(x)=\sqrt{\tan x-\tan ^2 x} \Rightarrow \tan x-\tan ^2 x \geq 0$

or $0 \leq \tan x \leq 1 \quad$ or $\quad x \in \bigcup_{n \in I}\left[n \pi, n \pi+\frac{\pi}{4}\right]$

(vi) $f(x)=\frac{1}{2 \sin \frac{x}{2} \mid} \quad \Rightarrow \quad \sin \frac{x}{2} \neq 0 \quad$ or $\quad x \neq 2 n \pi$

(vii) $f(x)=\sqrt{\log _{1 / 4}\left(\frac{5 x-x^2}{4}\right)} \Rightarrow \quad \frac{5 x-x^2}{4} \leq 1$ and $5 x-x^2>0$ or $\quad x \in(0,1] \cup[4,5)$

(viii) $f(x)=\log _{10}\left(1-\log _{10}\left(x^2-5 x+16\right)\right)  \Rightarrow \quad 1-\log _{10}\left(x^2-5 x+16\right)>0 \\$

$\text { or } \quad x^2-5 x+6<0  \text { or } \quad x \in(2,3)$