A function f defined by `f(x)=In(sqrt(x^(2)+1-x))` is

Options

A. an even function
B. an odd function
C. Both even and odd function
D. Neither even nor odd function

Explanation

To determine the nature of the function \( f(x) = \ln(\sqrt{x^2 + 1 - x}) \), we will check if it is an even function, odd function, or neither.

### Step-by-Step Solution:

1. **Define the function**:
\[ f(x) = \ln(\sqrt{x^2 + 1 - x}) \]

2. **Evaluate \( f(-x) \)**:
We need to find \( f(-x) \):
\[ f(-x) = \ln(\sqrt{(-x)^2 + 1 - (-x)}) = \ln(\sqrt{x^2 + 1 + x}) \]

3. **Compare \( f(-x) \) with \( f(x) \)**:
We need to see if \( f(-x) \) is equal to \( f(x) \) or \( -f(x) \):
- We have:
\[ f(x) = \ln(\sqrt{x^2 + 1 - x}) \]
- And:
\[ f(-x) = \ln(\sqrt{x^2 + 1 + x}) \]

4. **Check if \( f(-x) = f(x) \)** (even function):
For \( f(-x) \) to be equal to \( f(x) \):
\[ \ln(\sqrt{x^2 + 1 + x}) \stackrel{?}{=} \ln(\sqrt{x^2 + 1 - x}) \]
This is not true in general, so \( f(-x) \neq f(x) \).

5. **Check if \( f(-x) = -f(x) \)** (odd function):
For \( f(-x) \) to be equal to \( -f(x) \):
\[ \ln(\sqrt{x^2 + 1 + x}) \stackrel{?}{=} -\ln(\sqrt{x^2 + 1 - x}) \]
This can be rewritten as:
\[ \ln(\sqrt{x^2 + 1 + x}) = \ln\left(\frac{1}{\sqrt{x^2 + 1 - x}}\right) \]
Which simplifies to:
\[ \sqrt{x^2 + 1 + x} = \frac{1}{\sqrt{x^2 + 1 - x}} \]
This equality does not hold for all \( x \).

6. **Conclusion**:
Since \( f(-x) \neq f(x) \) and \( f(-x) \neq -f(x) \), we conclude that the function \( f(x) \) is neither even nor odd.

### Final Answer:
The function \( f(x) = \ln(\sqrt{x^2 + 1 - x}) \) is neither even nor odd.