consider `f:R-{0}toR` defined by `f(x)=1-e^((1)/(x)-1)`
Q. Which of the following is/are correct?

Options

A. `f(x)` is an increasing function
B. `f(x)` is decreasing function
C. `f(x)` is an odd function
D. `f(x)` is neither even nor odd function

Explanation

To solve the problem, we need to analyze the function \( f(x) = 1 - e^{\left(\frac{1}{x} - 1\right)} \) and determine which statements about it are correct.

### Step 1: Differentiate the function
To check if the function is increasing or decreasing, we first need to find its derivative \( f'(x) \).

\[ f'(x) = \frac{d}{dx} \left( 1 - e^{\left(\frac{1}{x} - 1\right)} \right) \]

Using the chain rule, we differentiate \( e^{u} \) where \( u = \frac{1}{x} - 1 \):

\[ f'(x) = -e^{\left(\frac{1}{x} - 1\right)} \cdot \frac{d}{dx}\left(\frac{1}{x} - 1\right) \]

Now, we differentiate \( \frac{1}{x} \):

\[ \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \]

Thus, we have:

\[ f'(x) = -e^{\left(\frac{1}{x} - 1\right)} \cdot \left(-\frac{1}{x^2}\right) = \frac{e^{\left(\frac{1}{x} - 1\right)}}{x^2} \]

### Step 2: Analyze the sign of the derivative
Now we need to analyze \( f'(x) \):

1. The term \( e^{\left(\frac{1}{x} - 1\right)} \) is always positive for all \( x \neq 0 \).
2. The term \( x^2 \) is also always positive for all \( x \neq 0 \).

Therefore, \( f'(x) > 0 \) for all \( x \neq 0 \). This means that the function \( f(x) \) is increasing for all \( x \neq 0 \).

### Step 3: Check if the function is odd or even
To check if \( f(x) \) is odd or even, we need to evaluate \( f(-x) \):

\[ f(-x) = 1 - e^{\left(\frac{1}{-x} - 1\right)} = 1 - e^{\left(-\frac{1}{x} - 1\right)} = 1 - \frac{1}{e} e^{-\frac{1}{x}} \]

Now, we need to check if \( f(-x) = -f(x) \) or \( f(-x) = f(x) \):

1. For \( f(-x) = -f(x) \):
\[ 1 - e^{\left(-\frac{1}{x} - 1\right)} \neq -\left(1 - e^{\left(\frac{1}{x} - 1\right)}\right) \]

2. For \( f(-x) = f(x) \):
\[ 1 - e^{\left(-\frac{1}{x} - 1\right)} \neq 1 - e^{\left(\frac{1}{x} - 1\right)} \]

Thus, \( f(x) \) is neither odd nor even.

### Conclusion
From our analysis, we conclude that:
- The function \( f(x) \) is increasing for all \( x \neq 0 \).
- The function is neither odd nor even.

### Final Answer
The correct options are:
- Option 1: Correct (function is increasing)
- Option 4: Correct (function is neither even nor odd)