Function $\cos ^{-1} \left(\log _{2}\left(x^{2}+5 x+8\right)\right)$
The function should exist when
$-1 \leq \log _{2}\left(x^{2}+5 x+8\right) \leq 1$
We take only, $\log _{2}\left(x^{2}+5 x+8\right) \leq 1$
$\Rightarrow \, x^{2}+5 x+8 \leq(2)^{1}$
$\left\{\because-1 \leq \log _{2}\left(x^{2}+5 x+8\right)\right.$
given imaginary values $\}$
$\Rightarrow \, x^{2}+5 x+6 \leq \,0$
$\Rightarrow \, (x+2)(x+3) \leq\, 0$
$\Rightarrow \, x \in[-3,-2]$
Hence, the domain of the function is $-3 \leq\, x \leq\,-2$
