Let `f (x)` be invertible function and let `f ^(-1) (x)` be is inverse. Let equation `f (f ^(-1) (x)) =f ^(-1)(x)` has two real roots `alpha and beta` (with in domain of `f(x)),` then :

Options

A. `f (x) =x ` also have same two rreal roots
B. `f ^(-1) (x)=x` also have same two real roots
C. `f (x)=f^(-1)(x)` also have same two real roots
D. Area of triangle formed by `(0,0), (alpha, f (alpha)), and (beta, f (beta))` is 1 unit

Answer

To solve the problem step by step, we need to analyze the given equation and its implications regarding the roots α and β.

### Step 1: Understand the Given Equation
The equation we are given is:
\[ f(f^{-1}(x)) = f^{-1}(x) \]
Since \( f(f^{-1}(x)) = x \) for an invertible function, we can rewrite the equation as:
\[ x = f^{-1}(x) \]

### Step 2: Rearranging the Equation
Rearranging the equation gives:
\[ f^{-1}(x) - x = 0 \]
This means we need to find the values of \( x \) for which this equation holds.

### Step 3: Finding Roots
We know that the equation has two real roots, α and β. This implies that the function \( f^{-1}(x) - x \) intersects the x-axis at two points.

### Step 4: Analyzing the Function
To analyze the function further, we can consider a specific example of \( f(x) \). Let's assume:
\[ f(x) = x^2 \]
Then, the inverse function would be:
\[ f^{-1}(x) = \sqrt{x} \]

### Step 5: Substitute into the Equation
Now substituting \( f^{-1}(x) \) into our rearranged equation:
\[ \sqrt{x} - x = 0 \]

### Step 6: Solving the Equation
To solve \( \sqrt{x} - x = 0 \), we can rearrange it as:
\[ \sqrt{x} = x \]
Squaring both sides gives:
\[ x = x^2 \]
Rearranging this leads to:
\[ x^2 - x = 0 \]
Factoring out \( x \):
\[ x(x - 1) = 0 \]

### Step 7: Finding the Roots
From the factored equation, we find:
1. \( x = 0 \) (which we can denote as α)
2. \( x = 1 \) (which we can denote as β)

Thus, the roots α and β are 0 and 1, respectively.

### Step 8: Verifying the Options
Now we need to verify the given options based on the roots we found:

1. **Option 1:** \( f(x) = x \)
- This leads to \( x - x = 0 \), which gives the roots \( x = 0 \) and \( x = 1 \). **Correct.**

2. **Option 2:** \( f^{-1}(x) = x \)
- This leads to \( \sqrt{x} - x = 0 \), which also gives the roots \( x = 0 \) and \( x = 1 \). **Correct.**

3. **Option 3:** \( f(x) = f^{-1}(x) \)
- This leads to \( x^2 - \sqrt{x} = 0 \), which again gives the roots \( x = 0 \) and \( x = 1 \). **Correct.**

4. **Option 4:** Area of triangle condition
- The area calculated from the points (0,0) and (1,1) gives an area of 0, which does not satisfy the condition of being 1. **Incorrect.**

### Conclusion
Thus, the correct options are 1, 2, and 3, while option 4 is incorrect.

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