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Options

A. 0
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the number of solutions to the equation \( f(x) - f^{-1}(x) = 0 \), where the function \( f(x) \) is defined in two cases:

1. \( f(x) = -x + 1 \) for \( x \leq 0 \)
2. \( f(x) = -(x - 1)^2 \) for \( x \geq 1 \)

### Step 1: Find the inverse function \( f^{-1}(x) \)

#### Case 1: \( x \leq 0 \)

For \( x \leq 0 \):
- Let \( y = f(x) = -x + 1 \)
- Rearranging gives \( x = 1 - y \)
- This is valid for \( x \leq 0 \), which means \( 1 - y \leq 0 \) or \( y \geq 1 \)

Thus, the inverse function in this case is:
\[ f^{-1}(y) = 1 - y \quad \text{for } y \geq 1 \]

#### Case 2: \( x \geq 1 \)

For \( x \geq 1 \):
- Let \( y = f(x) = -(x - 1)^2 \)
- Rearranging gives \( -(x - 1)^2 = y \) or \( (x - 1)^2 = -y \)
- Taking the square root gives \( x - 1 = \sqrt{-y} \) or \( x = \sqrt{-y} + 1 \)
- This is valid for \( x \geq 1 \), which means \( \sqrt{-y} + 1 \geq 1 \) or \( -y \geq 0 \) or \( y \leq 0 \)

Thus, the inverse function in this case is:
\[ f^{-1}(y) = \sqrt{-y} + 1 \quad \text{for } y \leq 0 \]

### Step 2: Set up the equation \( f(x) - f^{-1}(x) = 0 \)

Now we need to solve \( f(x) = f^{-1}(x) \) for both cases.

#### Case 1: \( x \leq 0 \)

Here, we have:
\[ f(x) = -x + 1 \quad \text{and} \quad f^{-1}(x) = 1 - x \]
Setting them equal gives:
\[ -x + 1 = 1 - x \]
This simplifies to:
\[ -x + 1 = 1 - x \implies 0 = 0 \]
This equation is always true, so every \( x \leq 0 \) is a solution. Thus, there are infinitely many solutions in this case.

#### Case 2: \( x \geq 1 \)

Here, we have:
\[ f(x) = -(x - 1)^2 \quad \text{and} \quad f^{-1}(x) = 1 - x \]
Setting them equal gives:
\[ -(x - 1)^2 = 1 - x \]
Expanding and rearranging:
\[ -(x^2 - 2x + 1) = 1 - x \implies -x^2 + 2x - 1 = 1 - x \]
\[ -x^2 + 3x - 2 = 0 \]
Multiplying by -1 gives:
\[ x^2 - 3x + 2 = 0 \]
Factoring:
\[ (x - 1)(x - 2) = 0 \]
Thus, the solutions are \( x = 1 \) and \( x = 2 \).

### Step 3: Count the total number of solutions

- From Case 1 (\( x \leq 0 \)): Infinitely many solutions
- From Case 2 (\( x \geq 1 \)): 2 solutions \( (x = 1 \text{ and } x = 2) \)

### Conclusion

The total number of solutions to the equation \( f(x) - f^{-1}(x) = 0 \) is infinite due to the solutions in Case 1. Therefore, the answer is:
\[ \text{The number of solutions is infinite.} \]