Let `f: [1,2] -> [1, 4] and g : [1,2] -> [2, 7]` be two continuous bijective functions such that `f(1)\=4 and g (2)=7`. Number ofsolution of the equation `f(x)=g(x)` in `(1,2)` is equal to

Answer

To solve the problem, we need to determine the number of solutions to the equation \( f(x) = g(x) \) in the interval \( (1, 2) \) given the properties of the functions \( f \) and \( g \).

### Step-by-Step Solution:

1. **Understand the Functions**:
- We know that \( f: [1, 2] \to [1, 4] \) and \( g: [1, 2] \to [2, 7] \).
- Both functions are continuous and bijective (one-to-one and onto).

2. **Determine the Range of Each Function**:
- Since \( f \) is a bijective function from \( [1, 2] \) to \( [1, 4] \), its range is exactly \( [1, 4] \).
- Similarly, since \( g \) is a bijective function from \( [1, 2] \) to \( [2, 7] \), its range is exactly \( [2, 7] \).

3. **Evaluate the Endpoints**:
- For \( f \):
- \( f(1) = 4 \) (given)
- \( f(2) \) must be in \( [1, 4] \) and since \( f \) is continuous and bijective, \( f(2) < 4 \).
- For \( g \):
- \( g(1) \) must be in \( [2, 7] \) and since \( g \) is continuous and bijective, \( g(1) > 2 \).
- \( g(2) = 7 \) (given).

4. **Graphical Representation**:
- Plot the range of \( f \) from \( 1 \) to \( 4 \) and the range of \( g \) from \( 2 \) to \( 7 \).
- The function \( f(x) \) starts at \( (1, 4) \) and ends below \( (2, 4) \).
- The function \( g(x) \) starts above \( (1, 2) \) and ends at \( (2, 7) \).

5. **Finding Intersections**:
- Since \( f(1) = 4 \) and \( g(1) > 2 \), \( f(1) > g(1) \).
- At \( x = 2 \), \( f(2) < 4 \) and \( g(2) = 7 \), thus \( f(2) < g(2) \).
- By the Intermediate Value Theorem, since \( f(x) \) is continuous and starts above \( g(x) \) at \( x=1 \) and ends below \( g(x) \) at \( x=2 \), there must be at least one point where \( f(x) = g(x) \) in the interval \( (1, 2) \).

6. **Conclusion**:
- Since both functions are continuous and bijective, and they cross exactly once in the interval \( (1, 2) \), we conclude that there is exactly **one solution** to the equation \( f(x) = g(x) \) in the interval \( (1, 2) \).

### Final Answer:
The number of solutions of the equation \( f(x) = g(x) \) in \( (1, 2) \) is **1**.