Statement-1 : Let `f : [1, oo) rarr [1, oo)` be a function such that `f(x) = x^(x)` then the function is an invertible function.
Statement-2 : The bijective functions are always invertible .

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement -1 is False, Statement -2 is False
D. Statement -1 is False, Statement -2 is True

Answer

To determine whether the function \( f(x) = x^x \) defined on the interval \([1, \infty)\) is invertible, we need to check if it is a bijective function, meaning it is both one-to-one (injective) and onto (surjective).

### Step 1: Check if the function is one-to-one (injective)

To check if \( f(x) \) is one-to-one, we can analyze its derivative.

1. Start with the function:
\[ y = f(x) = x^x \]

2. Take the natural logarithm of both sides:
\[ \log y = x \log x \]

3. Differentiate both sides with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^x) = x^x \left( \log x + 1 \right) \]

4. Now, analyze the derivative \( \frac{dy}{dx} \):
- For \( x \geq 1 \), \( \log x \geq 0 \), hence \( \log x + 1 > 0 \).
- Therefore, \( \frac{dy}{dx} > 0 \) for all \( x \geq 1 \).

Since the derivative is positive, \( f(x) \) is an increasing function, which implies that it is one-to-one.

### Step 2: Check if the function is onto (surjective)

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

1. Calculate \( f(1) \):
\[ f(1) = 1^1 = 1 \]

2. Calculate the limit as \( x \) approaches infinity:
\[ \lim_{x \to \infty} f(x) = \lim_{x \to \infty} x^x = \infty \]

3. Since \( f(x) \) is continuous and increasing from \( 1 \) to \( \infty \), the range of \( f(x) \) is \( [1, \infty) \).

4. The co-domain of \( f(x) \) is also \( [1, \infty) \).

Since the range equals the co-domain, \( f(x) \) is onto.

### Conclusion

Since \( f(x) = x^x \) is both one-to-one and onto, we conclude that \( f(x) \) is invertible.

### Statement Validation

- **Statement 1**: The function \( f(x) = x^x \) is an invertible function. **True**.
- **Statement 2**: Bijective functions are always invertible. **True**.

Both statements are correct.