Why are inverse functions reflections across y = x?

[pan90]

Can anyone explain the logic behind the answer?

View attachment 9221

Taken from HiSet free practice test

 

[MarkFL]

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:

$$y=x$$