Problem
ALG-B3-M02-P003 Constant shift
#3
★★☆☆☆ Level 2 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)-5\), and \(f(1)=8\). Find \(f\).
Hint. Put \(g(x)=f(x)-5\).
Then \(g(x+y)=g(x)+g(y)\), and \(g(1)=3\). On \(\mathbb Q\), \(g(q)=3q\). Hence \(f(q)=3q+5\). Verification is direct.
Goal: choose the right first substitution and then verify the found function.