Let `A={x""inR|xge0|`. A function `f:AtoA` is defined by `f(x)=x^(2)`. Which one of the following is correct ?

Options

A. The function does not have inverse
B. f is its own inverse
C. The function has an inverse but f is not its own inverse
D. None of the above

Explanation

To solve the problem, we need to analyze the function \( f: A \to A \) defined by \( f(x) = x^2 \) where \( A = \{ x \in \mathbb{R} | x \geq 0 \} \). We will determine the properties of this function, particularly focusing on its inverse.

### Step-by-step Solution:

1. **Define the Function**:
The function is defined as \( f(x) = x^2 \) for \( x \geq 0 \).

2. **Determine the Range of the Function**:
Since \( x \) is non-negative, the output of \( f(x) \) will also be non-negative. Thus, the range of \( f \) is \( [0, \infty) \).

3. **Find the Inverse Function**:
To find the inverse, we start with the equation \( y = f(x) = x^2 \). We need to express \( x \) in terms of \( y \):
\[ y = x^2 \implies x = \sqrt{y} \]
Since \( x \) is non-negative, we only consider the positive square root. Therefore, the inverse function is:
\[ f^{-1}(y) = \sqrt{y} \]

4. **Express the Inverse Function in Terms of \( x \)**:
We can replace \( y \) with \( x \) to express the inverse function as:
\[ f^{-1}(x) = \sqrt{x} \]

5. **Compare the Function and Its Inverse**:
We have:
- \( f(x) = x^2 \)
- \( f^{-1}(x) = \sqrt{x} \)

Clearly, \( f(x) \) and \( f^{-1}(x) \) are not the same function.

6. **Conclusion**:
The function \( f(x) = x^2 \) has an inverse \( f^{-1}(x) = \sqrt{x} \), and they are not equal to each other. Therefore, the correct option is that the function has an inverse, but it is not the same as the original function.

### Final Answer:
The correct option is that the function has an inverse, but it is not the same as the original function.

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