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: Isolate
\(e^{f(x)} = e - e^x\).

Step 2: Positivity
An exponential is always positive, so we need \(e - e^x > 0\), that is \(e^x < e\).

Step 3: Solve
Taking logarithm gives \(x < 1\). So the domain is \((-\infty,1)\), option (B).

Step 4: Why not the others
For \(x\ge1\), \(e-e^x \le 0\), so \(f(x)\) does not exist. So the domain cannot include 1 or larger values.

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