Problem
ALG-B1-M08-P009 Linear solutions with an extra term
#9
★★☆☆☆ Level 2 of 5
Find all linear functions \(f(x)=ax+b\) satisfying \(f(x+y)=f(x)+f(y)+5\) for all real \(x,y\).
Compare the constant terms after substitution.
Substitution gives \(a(x+y)+b=ax+b+ay+b+5\). The coefficients of \(x,y\) match. For constants, \(b=2b+5\), so \(b=-5\). Answer: \(f(x)=ax-5\), where \(a\) is arbitrary.
Testing a linear hypothesis without unnecessary complication.