If the domain of the function f(x)= ^-1((5-x)/(3+2x))+1 _e(10-x) is (-, ] [ , )- , then 6(…

Question

If the domain of the function $f(x)={\sin }^{-1}(\frac{5-x}{3+2x})+\frac{1}{{\log }_{e}(10-x)}$ is $(-\infty,\alpha ]\cup [\beta ,\gamma )-{\delta }$, then $6(\alpha +\beta +\gamma +\delta )$ is equal to

Accepted Answer

$\begin{gathered} f(x)={\sin }^{-1}[\frac{5-x}{3+2x}]+\frac{1}{{\log }_{e}(10-x)} \ -1\le \frac{5-x}{3+2x}\le 1 \ \frac{2-3x}{3+2x}\le 0 \ (3x-2)(2x+3)\ge 0 \ (3x-2)(2x+3)\ge 0x\in (-\infty,-\frac{3}{2}]\cup [\frac{2}{3},\infty) \ \frac{8+x}{3+2x}\ge 0 \ x\in (-\infty,-8]\cup [\frac{-3}{2},\infty) \ x\in (-\infty,-8]\cup [\frac{2}{3},10)-{9}=6(\alpha +\beta +\gamma +\delta )=70 \end{gathered}$

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