Problem
ALG-B3-M05-P009 Midpoints with Two Values
#9
★★★☆☆ Level 3 of 5
Find continuous \(f:\mathbb R\to\mathbb R\) if \(f(x+y)+f(x-y)=2f(x)\), \(f(0)=3\), and \(f(2)=9\).
The midpoint equation gives an affine function.
As in the theory, continuous solutions have the form \(f(x)=ax+b\). From \(f(0)=3\), we get \(b=3\). From \(f(2)=9\), \(2a+3=9\), so \(a=3\). The answer is \(f(x)=3x+3\).
A standard application of the Jensen-type equation.