Problem 5

Determine all functions

f:\mathbb R\to\mathbb R

such that (1)

f

is strictly increasing, and (2)

f(x)+g(x)=2x

for every real

x

, where

g

is the composition inverse of

f

Step 3 of 5: Obtain the analogous identity for the inverse

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g_n(x)-x=n(g(x)-x)=-n(f(x)-x)

Detailed analysis

The inverse $g$ is also strictly increasing and its inverse is $f$ . Applying the same argument to $g$ gives $g_n(x)-x=n(g(x)-x)$ . Since the original equation gives $g(x)-x=-(f(x)-x)$ , this is $g_n(x)-x=-n(f(x)-x)$ . The same identity holds with $x$ replaced by any input.