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 2 of 5: Sum the constant successive differences

Previous step Next step

f_n(x)-f_{n-1}(x)=f(x)-x\Rightarrow f_n(x)-x=n(f(x)-x)

Detailed analysis

The recurrence says consecutive differences are equal: $f_{n+2}(x)-f_{n+1}(x)=f_{n+1}(x)-f_n(x)$ . Starting with $f_1(x)-f_0(x)=f(x)-x$ and summing $n$ equal differences gives $f_n(x)-x=n(f(x)-x)$ .