Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& f_1(x)={\log }_5\left(18x-x^2-77\right)\\ & \therefore 18x-x^2-77>0\\ & x^2-18x+77<0\\ & x\in (7,11)\\ & \alpha =7,\beta =11\\ & f_2(x)={\log }_{(x-1)}\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)\\ & x>1,x-1\ne 1,\frac{2x^2+3x-2}{x^2-3x-4}>0\\ & x>1,x\ne 2,\frac{(2x-1)(x+2)}{(x-4)(x+1)}>0\end{array}$

$\begin{array}{rl}& \therefore x\in (4,\infty )\\ & \therefore \gamma =4\\ & \therefore {\alpha }^2+{\beta }^2+{\gamma }^2=49+121+16\\ & =186\end{array}$