Mathematics - Functions Question with Solution | TestHub

Given,

Function $f\left(x\right)=\frac{{\log }_{\left(x+1\right)}\left(x-2\right)}{e^{2{\log }_{e}x}-\left(2x+3\right)}, x\in R$

To find domain we will take following cases:

Case-1:

$x-2>0\Rightarrow x>2$

$x\in \left(2,\infty \right)$

Case-2:

$x+1>0\Rightarrow x>-1$

And, $x+1\ne 1\Rightarrow x\ne 0$

$x\in \left(-1,0\right)\cup \left(0,\infty \right)$

Case-3:

$x>0$

$x\in \left(0,\infty \right)$

Case-4:

$e^{2{\log }_e\left(x\right)}-\left(2x+3\right)\Rightarrow x^2-2x-3\ne 0$

$\Rightarrow \left(x-3\right)\left(x+1\right)\ne 0$

$x\ne -1,3$

From $\text{case }\left(1\right) \cap \text{case }\left(2\right) \cap \text{case }\left(3\right) \cap \text{case }\left(4\right)$ we get,

 $x\in \left(2,\infty \right)-\left\{3\right\}$