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 \( f(x) \) given the equation \( e^x + e^{f(x)} = e \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ e^x + e^{f(x)} = e \]
We can rearrange this to isolate \( e^{f(x)} \):
\[ e^{f(x)} = e - e^x \]

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

### Step 3: Finding the Domain
The expression \( \ln(e - e^x) \) is defined only when its argument is greater than zero:
\[ e - e^x > 0 \]
This simplifies to:
\[ e > e^x \]

### Step 4: Solving the Inequality
To solve the inequality \( e > e^x \), we can divide both sides by \( e \) (since \( e > 0 \)):
\[ 1 > e^{x-1} \]
Taking the natural logarithm of both sides gives:
\[ \ln(1) > x - 1 \]
Since \( \ln(1) = 0 \), we have:
\[ 0 > x - 1 \]
This simplifies to:
\[ x < 1 \]

### Step 5: Conclusion on the Domain
Since there are no restrictions on \( x \) from below, the domain of \( f(x) \) is:
\[ (-\infty, 1) \]

### Final Answer
Thus, the domain of \( f(x) \) is:
\[ (-\infty, 1) \]

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