MathLabs

Problem 5

Find all functions f:R→Rf:\mathbb R\to\mathbb R such that (f(x)+f(z))(f(y)+f(t))=f(xy−zt)+f(xt+yz)(f(x)+f(z))(f(y)+f(t))=f(xy-zt)+f(xt+yz) for all real numbers x,y,z,tx,y,z,t.
Step 2 of 3: Derive multiplicativity
Previous stepNext step
In plain words

Derive multiplicativity

f(xy)=f(x)f(y)f(xy)=f(x)f(y)
Detailed analysis

For a nonconstant solution, substitutions with zeros give f(0)=0f(0)=0; the other possibility makes ff constant. Further substitutions give multiplicativity and, by comparing signs, f(−x)=f(x)f(-x)=f(x). Thus f(x)≥0f(x)\ge0 for x≥0x\ge0 and f(1)=1f(1)=1.