Generalization arg max for a value for a function of the g(max(f)) or preimage of g?

Question

I want to refer to the argument value $x_o$ in which the function takes a given value, $f(x) = 0.5\ \text{max}_x(f(x))$.

Note that this value is a function of the max, $g(y) = 0.5\ y$

Can I use a generalization of arg max, arg min etc as?

$x_o = \text{arg}\ 0.5\ \text{max}_x (f(x))$

Or do I need to use the preimage as

$x_o = f^{-1}[g(\text{max}_x (f(x)))] = f^{-1}[0.5\ \text{max}_x (f(x))]$

Or is there any other way to do it?