Let `f: A to B` be a function such that `f (x)= sqrt(x-2)+sqrt(4-x,)` is invertible, then which of the following is not possible ?

Options

A. `A=[3,4]`
B. `A=[2,3]`
C. `A=[2,2sqrt3]`
D. `{2,2sqrt2]`

Answer

To solve the problem, we need to analyze the function \( f(x) = \sqrt{x-2} + \sqrt{4-x} \) and determine its domain and range, and then check which of the given options is not possible for an invertible function.

### Step 1: Determine the Domain of \( f(x) \)

The function \( f(x) \) involves square roots, which means we need to ensure that the expressions inside the square roots are non-negative.

1. **Condition from \( \sqrt{x-2} \)**:
\[ x - 2 \geq 0 \implies x \geq 2 \]

2. **Condition from \( \sqrt{4-x} \)**:
\[ 4 - x \geq 0 \implies x \leq 4 \]

Combining these two conditions, we find:
\[ 2 \leq x \leq 4 \]
Thus, the domain of \( f(x) \) is \( [2, 4] \).

### Step 2: Determine the Range of \( f(x) \)

Next, we will find the range of \( f(x) \) by evaluating the function at the endpoints of the domain.

1. **Evaluate \( f(2) \)**:
\[ f(2) = \sqrt{2-2} + \sqrt{4-2} = 0 + \sqrt{2} = \sqrt{2} \]

2. **Evaluate \( f(4) \)**:
\[ f(4) = \sqrt{4-2} + \sqrt{4-4} = \sqrt{2} + 0 = \sqrt{2} \]

3. **Evaluate \( f(3) \)**:
\[ f(3) = \sqrt{3-2} + \sqrt{4-3} = 1 + 1 = 2 \]

Now we can summarize the values:
- At \( x = 2 \), \( f(2) = \sqrt{2} \)
- At \( x = 3 \), \( f(3) = 2 \)
- At \( x = 4 \), \( f(4) = \sqrt{2} \)

The function is increasing from \( \sqrt{2} \) to \( 2 \) and then decreasing back to \( \sqrt{2} \). Therefore, the range of \( f(x) \) is \( [\sqrt{2}, 2] \).

### Step 3: Check the Options

Given that \( f(x) \) is invertible, it must be one-to-one. We need to check which of the following options is not possible:

- **Option A**: \( (3, 4) \)
- **Option B**: \( (2, 3) \)
- **Option C**: \( (2, 2\sqrt{3}) \)

1. **For \( (3, 4) \)**: The function is one-to-one in this interval, so this is possible.
2. **For \( (2, 3) \)**: The function is also one-to-one in this interval, so this is possible.
3. **For \( (2, 2\sqrt{3}) \)**: Since \( 2\sqrt{3} \) is greater than \( 2 \) but less than \( 3 \), we need to check if there are multiple \( x \) values that yield the same \( y \) value.

Since \( 2\sqrt{3} \) falls within the range of \( f(x) \) and the function is not one-to-one in that interval (as it reaches the same \( y \) value twice), this option is not possible.

### Conclusion

Thus, the option that is not possible for the invertible function \( f(x) \) is:
\[ \text{Option C: } (2, 2\sqrt{3}) \]