I don't quite follow the example either. For a list (2, 4, 6), the function is applied to the first number and the "inverse" function (though it's really not) on the last number of the list, with an argument of b=2. I must be missing something in the argument -- I'm wondering, why the choice of b=2? And was the function not applied to the number 4?
Edit:
I believe the author was trying to show (albeit incorrectly) the generalization of an arithmetic mean using f(x) = ax + b, with a = 1 and b = 2 (not f(a,b) as stated; plus the inverse function was applied incorrectly at the element level). The f-mean (Mf) is defined as:
Mf(x₁,...,xₙ) = f⁻¹( (1/n) ∑ₖⁿ f(xₖ) )
where the function f is injective and continuous. Different choices of f result in different means. This wikipedia article provides relevant details on the f-mean.
https://en.wikipedia.org/wiki/Quasi-arithmetic_mean