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If A and B are the points of intersectio...

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

A

A and B necessarity lie on the line y=x

B

A and B must be coincident

C

slope of line AB may be -1

D

None of these above

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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 \( y = f(x) \) represents a function. - The function \( y = f^{-1}(x) \) represents the inverse of that function. 2. **Points of Intersection**: - The points of intersection \( A \) and \( B \) occur where \( f(x) = f^{-1}(x) \). - This means we are looking for values of \( x \) such that \( f(x) = x \). 3. **Analyzing the Graphs**: - The line \( y = x \) is the line of symmetry for the function and its inverse. - If a point \( (a, b) \) is on the line \( y = f(x) \), then the corresponding point on the inverse function will be \( (b, a) \). 4. **Slope of Line AB**: - The slope of the line connecting points \( A \) and \( B \) (where \( A \) and \( B \) are the intersection points) can be determined. - If \( A = (a, f(a)) \) and \( B = (b, f(b)) \), then the slope of line \( AB \) is given by: \[ \text{slope} = \frac{f(b) - f(a)}{b - a} \] - Since \( f(a) = f^{-1}(a) \) and \( f(b) = f^{-1}(b) \), we can infer that the slope of line \( AB \) will be -1 if \( f(x) \) is a function that is symmetric about the line \( y = x \). 5. **Conclusion**: - The points of intersection \( A \) and \( B \) will lie on the line \( y = x \) if \( f(x) \) is such that \( f(x) = x \) at those points. - The line connecting points \( A \) and \( B \) will have a slope of -1, indicating that the inverse function reflects across the line \( y = x \). ### Final Result: - The points of intersection \( A \) and \( B \) of the functions \( y = f(x) \) and \( y = f^{-1}(x) \) will satisfy the property that the slope of the line connecting these points is -1.
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