Animated Solution for Mathematics - Functions: If the domain of the function ${\sin }^{-1}\left(\frac{3x-22}{2x-19}\right)+{\log }_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha ,\beta ]$, then $3\alpha +10\beta$ is equal to:

To find the domain of the function $f(x)={\sin }^{-1}\left(\frac{3x-22}{2x-19}\right)+{\log }_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$, we must identify the intersection of the valid input regions for both the inverse sine and the logarithmic components.

The inverse sine function ${\sin }^{-1}(u)$ is defined only for $u\in [-1,1]$. This imposes the constraint:

We solve this by splitting it into two inequalities. First, consider $\frac{3x-22}{2x-19}+1\ge 0$: Using the Wavy Curve Method with critical points $x=8.2$ and $x=9.5$, we obtain $x\in (-\infty ,8.2]\cup (9.5,\infty )$.

Next, consider $\frac{3x-22}{2x-19}-1\le 0$: With critical points $x=3$ and $x=9.5$, the solution is $x\in [3,9.5)$.

Intersecting these two sets, the "safe zone" for the inverse sine is $x\in [3,8.2]$.

The natural logarithm ${\log }_e(u)$ requires $u>0$. Thus, we must satisfy:

Factoring the quadratic expressions, we get: The critical points are $x=-2,1,\frac{5}{3},5$. Applying the Wavy Curve Method, the inequality holds for $x\in (-\infty ,-2)\cup (1,\frac{5}{3})\cup (5,\infty )$.

We now find the intersection of the two domains: $[3,8.2]$ and $(-\infty ,-2)\cup (1,\frac{5}{3})\cup (5,\infty )$.

Comparing these intervals, the only region that satisfies both conditions is $(5,8.2]$. This corresponds to the domain $(\alpha ,\beta ]$, where $\alpha =5$ and $\beta =8.2=\frac{41}{5}$.

We are asked to compute $3\alpha +10\beta$:

The final result is 97.