A function is called one - one if each element of domain has a distinct image of co - domain or for any two or more the two elements of domain, function doesn't have same value. Otherwise function will be many - one. Function is called onto if co - domain = Range otherwise into. Function which is both one - one and onto, is called bijective. answer is defined only for bijective functions.
Let `f:[a, oo)rarr[1, oo)` be defined as `f(x)=2^(x(x-1))` be invertible, then the minimum value of a, is

Options

A. 0
B. 1
C. `1//2`
D. none of these

Answer

To determine the minimum value of \( a \) such that the function \( f(x) = 2^{x(x-1)} \) is bijective from the domain \( [a, \infty) \) to the range \( [1, \infty) \), we need to ensure that the function is both one-to-one (injective) and onto (surjective).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function is given as \( f(x) = 2^{x(x-1)} \). This function is defined for \( x \geq a \) and we want it to map to \( [1, \infty) \).

2. **Finding the Range of the Function**:
- The expression \( x(x-1) \) is a quadratic function that opens upwards. It has its vertex at \( x = \frac{1}{2} \).
- The value of \( x(x-1) \) at \( x = 1 \) is \( 1(1-1) = 0 \).
- For \( x < 1 \), \( x(x-1) < 0 \), and for \( x > 1 \), \( x(x-1) > 0 \).
- Therefore, \( f(x) \) will take the value \( f(1) = 2^0 = 1 \) and will increase without bound as \( x \) increases beyond 1.

3. **Ensuring the Function is One-to-One**:
- To ensure that \( f(x) \) is one-to-one, we need to check the derivative \( f'(x) \).
- The derivative \( f'(x) \) can be found using the chain rule:
\[ f'(x) = 2^{x(x-1)} \cdot \ln(2) \cdot (2x - 1) \]
- The function \( f'(x) \) will be positive when \( 2x - 1 > 0 \) or \( x > \frac{1}{2} \).
- Therefore, \( f(x) \) is increasing for \( x \geq \frac{1}{2} \).

4. **Determining the Minimum Value of \( a \)**:
- Since we want \( f(x) \) to be bijective, we need \( a \) to be at least 1 because \( f(1) = 1 \) is the minimum value of the range.
- If \( a < 1 \), then \( f(x) \) would not cover the entire range starting from 1.

5. **Conclusion**:
- The minimum value of \( a \) such that \( f(x) \) is bijective is \( a = 1 \).

### Final Answer:
The minimum value of \( a \) is \( \boxed{1} \).