Problem
ALG-B3-M05-P014 A Shift and a Product
#14
★★★★☆ Level 4 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(x+y)=f(x)+f(y)-1\) and \(f(xy)=f(x)f(y)-f(x)-f(y)+2\).
Put \(g(x)=f(x)-1\).
Let \(g(x)=f(x)-1\). The first condition gives \(g(x+y)=g(x)+g(y)\). The second condition, after substituting \(f=g+1\), becomes \(g(xy)=g(x)g(y)\). On \(\mathbb Q\), an additive function has the form \(g(x)=cx\), and multiplicativity gives \(c=0\) or \(c=1\). Thus \(f(x)=1\) or \(f(x)=x+1\). Both functions work.
A good task on signs during a shift.