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I'm solving an expression like so:

Solve[u[x] == u[a] + u[b], x]

Mathematica returns

{{x->(u^(-1))[u[a]+u[b]]}}

If I now specify the function u more precisely, via

% /. {u[x_] -> x}

what I get is

{{x->(u^(-1))[a+b]}}

i.e. Mathematica replaces the function but not its inverse. Is there any way of replacing the inverse appropriately as well?

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You need to replace function u with your definition, not only the u[x] symbol. Defining u as a function solves the problem

Solve[u[x] == u[a] + u[b], x] /. {u -> Function[x, x]}
(* ==>  {{x -> a + b}} *)
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