Question

Let the function $f$ be defined in $(0,1)$. Two more functions are given as $g ( x )= e ^{ x }$ and $h ( x )=\ln | x |$
Domain of $f ( g ( x ))+ f ( h ( x ))$ is

• A. $(-e,-1)$
• B. $(- e ,-1) \cup(1, e )$
• C. $[- e ,-1]$
• D. $(-\infty, 0)$

Answer 1

Answer: (A)

$f ( g ( x ))+ f ( h ( x ))= f \left( e ^{ x }\right)+ f (\ln | x |) $
$\because f \text { is defined in }(0,1) $
$\therefore 0< e ^{ x }<1 \text { and } 0<\ln | x |<1$
$ x \in(-\infty, 0) \text { and } 1<| x |< e $
$ x \in(- e ,-1) \cup(1, e ) $
$\therefore \text { Domian is }(- e ,-1)$