If the domain of the function f(x)= _e(2x+34x^2+x-3)+ ^-1((2x-1)/(x+2)) is ( , ], then the…

Question

If the domain of the function $f(x)={\log }_{e}(\frac{2x+3}{4{x}^{2}+x-3})+{\cos }^{-1}(\frac{2x-1}{x+2})$ is $(\alpha ,\beta ]$, then the value of $5\beta -4\alpha $ is equal to

Accepted Answer

$\begin{gathered} \frac{2x+3}{4{x}^{2}+x-3}>0-1\le \frac{2x-1}{x+2}\le 1 \ \frac{2x+3}{(4x-3)(x+1)}>0\, \frac{3x+1}{x+2}\ge 0\frac{x-3}{x+2}\le 0 \end{gathered}$

$(-\infty,-2)\cup [\frac{-1}{3},\infty)$ ..... (1)

$(-2,3]$ ..... (2)

$[\frac{-1}{3},3]$ ..... (3) $\, (1)\cap (2)\cap (3)$

$\begin{gathered} (\frac{3}{4},3] \ \alpha =\frac{3}{4}\beta =3 \ 5\beta -4\alpha =15-3=12 \end{gathered}$

Solutions · 1