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\(\displaystyle (x,f(x))\)

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

\(\displaystyle (f(x),x)\)

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

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

Thereby implying that the line of symmetry must be:

\(\displaystyle y=x\)