If D is the set of all real x such that `1-e^((1)/(x)-1)` is positive , then D is equal to

Options

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

Answer

To solve the problem, we need to determine the set \( D \) of all real numbers \( x \) such that the expression \( 1 - e^{\frac{1}{x} - 1} \) is positive.

### Step-by-step Solution:

1. **Set Up the Inequality**:
We start with the inequality:
\[ 1 - e^{\frac{1}{x} - 1} > 0 \]

2. **Rearranging the Inequality**:
We can rearrange this inequality to isolate the exponential term:
\[ e^{\frac{1}{x} - 1} < 1 \]

3. **Taking the Natural Logarithm**:
Since the exponential function is always positive, we can take the natural logarithm of both sides. However, we must remember that the logarithm of 1 is 0:
\[ \frac{1}{x} - 1 < 0 \]

4. **Solving the Inequality**:
Now, we can solve the inequality:
\[ \frac{1}{x} < 1 \]
This can be rewritten as:
\[ 1 < x \]
or
\[ x > 1 \]

5. **Conclusion**:
Therefore, the set \( D \) of all real numbers \( x \) such that \( 1 - e^{\frac{1}{x} - 1} \) is positive is:
\[ D = (1, \infty) \]