Question

If $f(x)$ is defined on $(0, 1)$, then the domain of definition of $f (e^x ) + f (ln | x |)$ is

• A. $(-e, -1)$
• B. $(-e, -1) \cup (1, e)$
• C. $\left(-\infty, -1\right) \cup \left(1, \infty\right)$
• D. $(-e, e)$

Answer 1

Answer: (A)

Since the domain of f is $(0, 1)$,
$\therefore \quad0 < e^{x} < 1$ and $0 < ln \left|x\right| < 1$
$\Rightarrow log \,0 < x< log \,1$ and $e^{0} < \left|x\right| < e^{1}$
$\Rightarrow -\infty < x < 0$ and $1 < \left|x\right| < e$
$\Rightarrow x \in\left( -\infty, 0\right)$ and $x \in \left( -e, -1\right) \cup \left(1, e\right)$
$\Rightarrow x \in \left( -e, -1\right)$