Question

If the domain of $f(x)$ is $(0,1)$, then the domain of $y=f(e^{x})+f(\ln|x|)$ is:

• A. $(-1,-\frac{1}{e})$
• B. $(\frac{1}{e},1)$
• C. $(-e,-1)$
• D. $(-e,-1)\cup(1,e)$

Hint:

For sums of functions like $$ f(g(x))+f(h(x)), $$ both functions must exist simultaneously. Always take the intersection of the individual domains.

Answer 1

The Correct Option is C

Concept: For a composite function $f(g(x))$ to be defined, the value of the inner function $g(x)$ must belong to the domain of $f$. Since the domain of $f(x)$ is $(0,1)$, every input given to $f$ must satisfy: $$ 0<\text{input}<1 $$ For the sum $$ y=f(e^x)+f(\ln|x|) $$ both terms must be defined simultaneously. Therefore, we find the domain of each term separately and then take their intersection. Step 1: Find the domain of $f(e^x)$.
Since the input of $f$ must lie in $(0,1)$: $$ 0<e^x<1 $$ Now, $$ e^x>0 \quad \text{for all real } x $$ and $$ e^x<1 \iff x<0 $$ Hence, $$ x\in(-\infty,0) \quad \cdots (1) $$

Step 2: Find the domain of $f(\ln|x|)$.
Again, the input of $f$ must lie in $(0,1)$: $$ 0<\ln|x|<1 $$ Exponentiating throughout: $$ e^0<|x|<e^1 $$ Thus, $$ 1<|x|<e $$ This gives two intervals: $$ x\in(-e,-1)\cup(1,e) \quad \cdots (2) $$

Step 3: Take the intersection of the domains.
The required domain is: $$ (-\infty,0)\cap\Big[(-e,-1)\cup(1,e)\Big] $$ Only the negative interval survives: $$ \boxed{(-e,-1)} $$ Hence, the correct answer is: $$ \boxed{(-e,-1)} $$ which matches option (C).