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 3 of 3: Normalize the monotone multiplicative function
Previous stepNext step
In plain words

Normalize the monotone multiplicative function

f(x)=x2f(x)=x^2
Detailed analysis

Putting (z,t)=(y,x)(z,t)=(y,x) gives f(x2+y2)=(f(x)+f(y))2f(x^2+y^2)=(f(x)+f(y))^2, so ff is weakly increasing on [0,∞)[0,\infty). A nonzero monotone multiplicative function is ∣x∣r|x|^r; the all-one substitution gives 2r=42^r=4, hence r=2r=2.