The function `f : R -> R` defined as `f (x) =1/2 In (sqrt(sqrt(x^2+1)+x) + sqrt(sqrt(x^2+1)-x))` is

Options

A. one-one and onto both
B. one-one but not onto
C. onto but not one-one
D. Neither one-one nor onto

Answer

To analyze the function \( f : \mathbb{R} \to \mathbb{R} \) defined as

\[ f(x) = \frac{1}{2} \ln \left( \sqrt{\sqrt{x^2+1}+x} + \sqrt{\sqrt{x^2+1}-x} \right) \]

we need to determine whether it is a one-one function and whether it is onto.

### Step 1: Check if the function is one-one

To check if the function is one-one, we will evaluate \( f(-x) \):

\[ f(-x) = \frac{1}{2} \ln \left( \sqrt{\sqrt{(-x)^2+1}+(-x)} + \sqrt{\sqrt{(-x)^2+1}-(-x)} \right) \]

This simplifies to:

\[ f(-x) = \frac{1}{2} \ln \left( \sqrt{\sqrt{x^2+1}-x} + \sqrt{\sqrt{x^2+1}+x} \right) \]

Notice that \( f(-x) \) is not equal to \( f(x) \) because the terms inside the logarithm are arranged differently. However, we can see that:

\[ f(x) = f(-x) \]

This indicates that the function is symmetric about the y-axis, which means it is not one-one.

### Step 2: Check if the function is onto

To check if the function is onto, we need to determine the range of \( f(x) \).

1. **Evaluate the expression inside the logarithm:**

\[ \sqrt{\sqrt{x^2+1}+x} + \sqrt{\sqrt{x^2+1}-x} \]

For all \( x \in \mathbb{R} \), both \( \sqrt{x^2+1} + x \) and \( \sqrt{x^2+1} - x \) are positive, thus the entire expression is positive.

2. **Determine the minimum value of \( f(x) \):**

As \( x \) approaches \( -\infty \) or \( +\infty \), we can analyze the limits:

- When \( x \to \infty \):
\[ f(x) \to \frac{1}{2} \ln(2x) \to \infty \]
- When \( x \to -\infty \):
\[ f(-x) \to \frac{1}{2} \ln(2(-x)) \to \infty \]

The function approaches infinity in both directions.

3. **Evaluate \( f(0) \):**
\[ f(0) = \frac{1}{2} \ln \left( \sqrt{1} + \sqrt{1} \right) = \frac{1}{2} \ln(2) \]

Since \( f(x) \) is always greater than \( 0 \) and approaches infinity, the range of \( f(x) \) is \( \left( \frac{1}{2} \ln(2), \infty \right) \).

Since the codomain is \( \mathbb{R} \) and the range does not cover all real numbers (specifically, it does not include negative values), the function is not onto.

### Conclusion

The function \( f(x) \) is neither one-one nor onto.

### Final Answer

The correct option is D: The function is neither one-one nor onto.

---