Question

If $f(x)$ is defined on $(0,1)$, then the domain of definition of $f(e^x)+f(lnâˆ\pounds xâˆ\pounds )$ is

• A. $(−e,−1)$
• B. $(−e,−1)∪(1,e)$
• C. $\left(−∞,−1\right)∪\left(1,∞\right)$
• D. $(−e,e)$

Answer 1

Answer: (A)

Since the domain of f is $(0,1)$,
$∴0<e^x<1$ and $0<lnâˆ\pounds xâˆ\pounds <1$
$⇒log0<x<log1$ and $e^0<âˆ\pounds xâˆ\pounds <e^1$
$⇒−∞<x<0$ and $1<âˆ\pounds xâˆ\pounds <e$
$⇒x∈\left(−∞,0\right)$ and $x∈\left(−e,−1\right)∪\left(1,e\right)$
$⇒x∈\left(−e,−1\right)$