If P and Q are the points of intersection of y=f(x) and y=`f^(-1)` (x), then

Options

A. P and Q necessarily lie on the line y = x
B. P and Q must be coincident
C. Slope of PQ may be -1
D. None

Answer

To solve the problem, we need to analyze the points of intersection of the functions \( y = f(x) \) and \( y = f^{-1}(x) \).

### Step-by-Step Solution:

1. **Understanding the Functions**:
The function \( f(x) \) and its inverse \( f^{-1}(x) \) are such that if \( (a, b) \) is a point on the graph of \( f(x) \), then \( (b, a) \) is a point on the graph of \( f^{-1}(x) \). This means that the points of intersection will satisfy both equations simultaneously.

2. **Setting the Equations Equal**:
To find the points of intersection, we set:
\[ f(x) = f^{-1}(x) \]
This implies that at the points of intersection, the output of the function is equal to the output of its inverse.

3. **Using the Identity of Inverse Functions**:
For any function \( f \) and its inverse \( f^{-1} \), the points where \( f(x) = f^{-1}(x) \) must also satisfy:
\[ f(f(x)) = x \]
This means that if \( y = f(x) \), then \( y \) must also equal \( x \) at the points of intersection.

4. **Identifying the Line of Intersection**:
The line \( y = x \) is significant here because it represents all points where the output of the function equals the input. Therefore, the points of intersection \( P \) and \( Q \) must lie on this line.

5. **Conclusion about Points of Intersection**:
Since \( f(x) = f^{-1}(x) \) at the points of intersection, and this occurs when \( x = y \), we conclude that:
- The points \( P \) and \( Q \) must lie on the line \( y = x \).
- Therefore, the intersection points \( P \) and \( Q \) are necessarily on the line \( y = x \).

### Final Answer:
The points of intersection \( P \) and \( Q \) of the functions \( y = f(x) \) and \( y = f^{-1}(x) \) lie on the line \( y = x \).