A given function $f(x)$ has an inverse $f^{-1}(x)$ - HSC - SSCE Mathematics Extension 1 - Question 9 - 2022 - Paper 1

To analyze the intersection of the graphs $y = f(x)$ and $y = f^{-1}(x)$, we consider the properties of inverse functions.

1. Common Point on the Line $y = x$: For any point $(a, b)$ on the graphs of both functions, the condition $f(a) = f^{-1}(a)$ must hold true. This indicates that at points of intersection, it is likely that $a = b$, thus these points could lie on the line $y = x$.

2. Tangent Slopes: The derivative of the inverse function $f^{-1}(x)$ is given by: (f^{-1})'(y) = rac{1}{f'(x)} At the points where these graphs intersect, by the Inverse Function Theorem, the slopes of the tangents will not be parallel or identical except at specific cases unless they are specifically constructed to be that way.

3. Perpendicularity of Tangents: For the tangents to be perpendicular at a point of intersection $(a, b)$, we must have: $f'(a) imes (f^{-1})'(b) = -1$ Given the earlier analysis and common behavior of inverse functions, this condition generally holds true due to the nature of their derivations.