Let `F:[3,infty]to[1,infty]` be defined by `f(x)=pi^(x(x-3)`, if `f^(-1)(x)` is inverse of `f(x)` then the number of solution of the equation `f(x)=f^(-1)(x)` are

Explanation

To solve the problem, we need to find the number of solutions to the equation \( f(x) = f^{-1}(x) \), where \( f(x) = \pi^{x(x-3)} \). We can use the property that \( f^{-1}(x) \) is the reflection of \( f(x) \) about the line \( y = x \). Therefore, the equation \( f(x) = f^{-1}(x) \) is equivalent to finding the points where \( f(x) = x \).

### Step-by-Step Solution:

1. **Define the function**: We have \( f(x) = \pi^{x(x-3)} \) for \( x \geq 3 \).

2. **Set up the equation**: We need to solve the equation \( f(x) = x \). This gives us:
\[ \pi^{x(x-3)} = x \]

3. **Analyze the behavior of \( f(x) \)**:
- At \( x = 3 \):
\[ f(3) = \pi^{3(3-3)} = \pi^0 = 1 \]
- As \( x \to \infty \), \( f(x) \) grows rapidly because the exponential function dominates.

4. **Check the monotonicity of \( f(x) \)**:
- To determine if \( f(x) \) is increasing, we can find its derivative \( f'(x) \). Since \( f(x) \) is an exponential function, it is strictly increasing for \( x \geq 3 \).

5. **Graphical interpretation**:
- The line \( y = x \) is a straight line passing through the origin with a slope of 1.
- The function \( f(x) \) starts at \( (3, 1) \) and increases without bound as \( x \) increases.

6. **Finding intersections**:
- Since \( f(x) \) is continuous and strictly increasing, and it starts below the line \( y = x \) at \( x = 3 \) (where \( f(3) = 1 < 3 \)), and eventually \( f(x) \) will surpass \( y = x \) as \( x \) increases.
- By the Intermediate Value Theorem, there must be exactly one point where \( f(x) = x \) since \( f(x) \) is continuous and strictly increasing.

7. **Conclusion**:
- Therefore, the number of solutions to the equation \( f(x) = f^{-1}(x) \) is **1**.

### Final Answer:
The number of solutions of the equation \( f(x) = f^{-1}(x) \) is **1**.