MathLabs

Problem 5

Find all functions f:R→Rf:\mathbb R\to\mathbb R such that for all x,y,z∈Rx,y,z\in\mathbb R, f(f(x)+f(y)+f(z))=f(f(x)−f(y))+f(2xy+f(z))+2f(xz−yz)f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).
Step 4 of 8: Evenness
Previous stepNext step
f(x)=f(−x) (∀x)f(x)=f(-x)\ (\forall x)
Detailed analysis

Choose s0s_0 with f(s0)≠f(0)f(s_0)\ne f(0) and apply the preceding implication to u1=f(s0)−f(0)u_1=f(s_0)-f(0) and u2=−u1u_2=-u_1. Since u1≠0u_1\ne0, this yields f(x)=f(−x)f(x)=f(-x).