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 8 of 8: Verify
Previous stepNext step
f≡0orf(x)=x2f\equiv0\quad\text{or}\quad f(x)=x^2
Detailed analysis

Direct substitution verifies both the zero function and f(x)=x2f(x)=x^2; the preceding argument proves that no others exist.