Why can we do this to find inverse functions?

One way to find the two-sided inverse of a function is to solve the equation f(x) = y for the variable x. But why does this always give us an inverse function? In this video, we give a proof and explanation of where this method comes from.
Functions playlist: • Functions
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Music: OcularNebula - The Lopez

[@I0MSammy]

You, Dr Trefor Bazett and Grant Sanderson are the best teachers of mathematics on yt.

[@pages.edits1]

very nice 👌

[@That_One_Guy...]

I would do it like this :
f(x) = 2x-3
replace x with x/2 :
f(x/2) = 2(x/2)-3 = x-3
replace x with x+3 :
f( (x+3)/2 ) = x => f^{-1}(x) = f^{-1}( f( (x+3)/2 ) ) = (x+3)/2
at least that make sense notationically.

[@mahmoudalbahar1641]

Thank you for your great videos...
And I want to ask you:
I found from wolfram alpha that
sum of k^(-k) from k=1 to k=infinity is: 1.29129 but why?