The discussion centers on the relationship between a function \(f(x)\) and its inverse \(g(x)\). It highlights that if a point \((x, f(x))\) exists on the graph of \(f(x)\), then the corresponding point \((f(x), x)\) will be on the graph of \(g(x)\). The conversation emphasizes the significance of the mid-point locus, which is \(\left(\frac{x+f(x)}{2}, \frac{x+f(x)}{2}\right)\). This leads to the conclusion that the line of symmetry for the functions is \(y = x\). Understanding this symmetry is crucial for grasping the concept of inverse functions.
