The domain of `f(x)` is, if `e^(x)+e^(f(x))=e`

Options

A. `(-oo,1)`
B. `(-oo,1]`
C. `(-oo,0]`
D. `(-oo,0)`

Answer

To find the domain of the function \( f(x) \) defined by the equation \( e^x + e^{f(x)} = e \), we can follow these steps:

### Step 1: Rearranging the Equation
Start by isolating \( e^{f(x)} \):
\[ e^{f(x)} = e - e^x \]

### Step 2: Taking the Natural Logarithm
Next, we take the natural logarithm of both sides:
\[ f(x) = \ln(e - e^x) \]

### Step 3: Finding the Domain
For the logarithm to be defined, the argument must be positive:
\[ e - e^x > 0 \]

### Step 4: Solving the Inequality
Rearranging the inequality gives:
\[ e > e^x \]
Dividing both sides by \( e \) (which is positive) results in:
\[ 1 > e^{x - 1} \]
Taking the natural logarithm of both sides:
\[ \ln(1) > x - 1 \]
Since \( \ln(1) = 0 \), we have:
\[ 0 > x - 1 \]
This simplifies to:
\[ x < 1 \]

### Step 5: Conclusion
Thus, the domain of \( f(x) \) is:
\[ (-\infty, 1) \]

### Summary of Steps:
1. Rearranged the original equation to isolate \( e^{f(x)} \).
2. Took the natural logarithm to express \( f(x) \).
3. Established the condition for the logarithm to be defined.
4. Solved the inequality to find the values of \( x \) that satisfy the condition.
5. Concluded the domain.