Let the domains of the functions f(x)= _4 _3 _7(8- _2(x^2+4 x+5)) and g(x)= ^-1((7 x+10)/(…

Question

Let the domains of the functions $f(x)=\log _4 \log _3 \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)$ and $\mathrm{g}(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)$ be $(\alpha, \beta)$ and $[\gamma, \delta]$, respectively. Then $\alpha^2+\beta^2+\gamma^2+\delta^2$ is equal to :

Accepted Answer

$f(x)=\log _4\left(\log _3\left(\log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)\right)\right.$ $$\begin{aligned} & \log _3\left(\log _1\left(8-\log _2\left(x^2+4 x+5\right)\right)\right)>0 \ & \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)>1 \ & 8-\log _2\left(x^2+4 x+5\right)>7 \ & -\log _2\left(x^2+4 x+5\right)>-1 \ & \log _2\left(x^2+4 x+5\right)<1 \ & x^2+4 x+5<2 \ & x^2+4 x+3<0 \ & \Rightarrow(x+3)(x+1)<0 \quad \ldots(1) \ & \log _7\left(8-\log _2\left(x^2+4 x+5\right)\right)>0 \ & 8-\log _2\left(x^2+4 x+5\right)>1 \ & \log _2\left(x^2+4 x+5\right)<9 \ & x^2+4 x+5<2^9 \ & x^2+4 x+5<512 \ & \Rightarrow x^2+4 x-507<0 \ & \Rightarrow x=-4 \pm \sqrt{16+2028} \end{aligned}$$ $$\begin{aligned} & x=\frac{-4 \pm \sqrt{2044}}{2} \quad\text{..... (2)}\ & \Rightarrow\left(x-\left(\frac{-4+\sqrt{2044}}{2}\right)\right)\left(x-\left(\frac{-4-\sqrt{2044}}{2}\right)\right)<0 \ & x^2+4 x+5>0 \ & D>0 \ & x \in R \ & \text { Also, } 8-\log _2\left(x^2+4 x+5\right)>0 \ & \log _2\left(x^2+4 x+5\right)<8 \ & x^2+4 x+5<256 \ & \Rightarrow x^2+4 x-251<0 \ & \Rightarrow x=-4 \pm \sqrt{16+1004} \ & \Rightarrow x=\frac{-4 \pm \sqrt{1020}}{2} \end{aligned}$$ $$\begin{aligned} &\Rightarrow\left(x-\left(\frac{-4+\sqrt{1020}}{2}\right)\right)\left(x-\left(\frac{-4-\sqrt{1020}}{2}\right)\right)<0\ &\therefore \text { Intersection of (1), (2) and (3) } \end{aligned}$$ $$\begin{aligned} & \therefore x \in(-3,-1) \ & -1 \leq \frac{7 x+10}{x-2} \leq 1 \ & \Rightarrow x \in[-2,-1] \ & \therefore \alpha^2+\beta^2+\gamma^2+\delta^2=(-3)^2+(-1)^2+(-2)^{-2}+(-1)^2 \ & =9+1+4+1 \ & =15 \end{aligned}$$

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