The domain of f(x) = _ (x + 1) (x - 2) e ^ 2 _ e x - (2x + 3) , x ∈...

Given,

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

To find domain we will take following cases:

Case-1:

$x - 2 > 0 ⇒ x > 2$

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

Case-2:

$x + 1 > 0 ⇒ x > - 1$

And, $x + 1 \ne 1 ⇒ x \ne 0$

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

Case-3:

$x > 0$

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

Case-4:

${e}^{2\log_{e}\left(x\right)} - \left(2x + 3\right) ⇒ {x}^{2} - 2x - 3 \ne 0$

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

$x \ne - 1, 3$

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

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