Problem
ALG-B1-M08-P014 Two linear relations
#14
★★★☆☆ Level 3 of 5
Find all linear functions \(f(x)=ax+b\) such that \(f(x)+f(1-x)=1\) and \(f(x+1)=f(x)+1\) for all \(x\).
The second condition first finds \(a\), the first finds \(b\).
From \(f(x+1)=f(x)+1\), we get \(a=1\). Thus \(f(x)=x+b\). The first condition gives \(x+b+1-x+b=1\), so \(1+2b=1\), hence \(b=0\). Answer: \(f(x)=x\).
A good problem on using two conditions together.