Question

The domain of f(x) is (0, 1). Then the domain of $f\left(e^x\right)+f\left(ln|x|\right)$ is

• A. $(-1,e)$
• B. $(1,e)$
• C. $(-e,-1)$
• D. $(-e,1)$

Answer 1

The correct option is C $(-e,-1)$

To find the domain of $f\left(e^x\right)+f\left(ln|x|\right)$.
Given that the domain of f(x) is (0, 1)
$\Rightarrow 0<e^x<1and0<ln|x|<10<e^x<1\Rightarrow xϵ(-\infty ,0).....\left(i\right)0<ln|x|<1\Rightarrow 1<|x|<e\Rightarrow xϵ(-e,-1)\cup (1,e)....\left(ii\right)$

Combining (i) and (ii), the domain of $f\left(e^x\right)+f\left(ln|x|\right)$ is (-e,-1)