Question

If \(e^x+e^{f(x)} = e\), then the domain of \(f(x)\) is

• A. \((1,\infty )\)
• B. \((-\infty ,1)\)
• C. \((-\infty ,\infty )\)
• D. \((-\infty ,0)\)

Hint:

Since e^f(x) must be positive, e^x must be less than e.

Answer 1

The Correct Option is B

Step 1: Take logarithm
$f(x) = \ln(e - e^x)$, which exists only when the argument is positive.

Step 2: Inequality
$e^x<e\Rightarrow x<1$. Option (B).

Final Answer:
(-infinity, 1). \[ \boxed{\text{(B)}\ (-\infty,1)} \]