MHB Inverse Functions: Reflection of f(x) & g(x) Logic

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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.
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Can anyone explain the logic behind the answer?

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Suppose we have the point:

$$(x,f(x))$$

on the plot of \(f(x)\). Then, on the plot of \(g(x)\), we must have the corresponding point:

$$(f(x),x)$$

Now, consider that for all possible points, the locus of the mid-points is:

$$\left(\frac{x+f(x)}{2},\frac{x+f(x)}{2}\right)$$

Thereby implying that the line of symmetry must be:

$$y=x$$
 

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