Cosider `f(x)=1-e^((1)/(x)-1)`
If S is the range of f(x), then S is

Options

A. `(-infty,infty)`
B. `(-infty,1)`
C. `(-infty,1)`
D. `(-infty,1)-{(1-(1)/(e))}`

Explanation

To find the range of the function \( f(x) = 1 - e^{\left(\frac{1}{x} - 1\right)} \), we will follow these steps:

### Step 1: Set \( f(x) = y \)
We start by letting \( y = f(x) \):
\[ y = 1 - e^{\left(\frac{1}{x} - 1\right)} \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ e^{\left(\frac{1}{x} - 1\right)} = 1 - y \]

### Step 3: Take the natural logarithm
Taking the natural logarithm of both sides:
\[ \frac{1}{x} - 1 = \ln(1 - y) \]

### Step 4: Solve for \( \frac{1}{x} \)
Rearranging gives:
\[ \frac{1}{x} = \ln(1 - y) + 1 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{1}{\ln(1 - y) + 1} \]

### Step 6: Determine the conditions for \( y \)
For \( x \) to be defined, the denominator must not be zero:
\[ \ln(1 - y) + 1 \neq 0 \implies \ln(1 - y) \neq -1 \implies 1 - y \neq \frac{1}{e} \implies y \neq 1 - \frac{1}{e} \]

### Step 7: Determine the range of \( y \)
Next, we need to consider the conditions under which \( \ln(1 - y) \) is defined:
\[ 1 - y > 0 \implies y < 1 \]

### Step 8: Combine the conditions
From the above steps, we have:
- \( y < 1 \)
- \( y \neq 1 - \frac{1}{e} \)

### Conclusion
Thus, the range \( S \) of the function \( f(x) \) is:
\[ S = (-\infty, 1) \setminus \left\{ 1 - \frac{1}{e} \right\} \]

### Final Answer
The range \( S \) is:
\[ S = (-\infty, 1) \text{ excluding } \left\{ 1 - \frac{1}{e} \right\} \]