MathLabs

Problem 5

Find all functions f:R→Rf:\mathbb R\to\mathbb R that are bounded above and satisfy f(xf(y))+yf(x)=xf(y)+f(xy)f(xf(y))+yf(x)=xf(y)+f(xy) for all real x,yx,y.
Step 3 of 7: Double the range
Previous stepNext step
f(f(y))=2f(y)f(f(y))=2f(y)
Detailed analysis

Putting x=1x=1 yields f(f(y))=2f(y)f(f(y))=2f(y). Thus if tt lies in the range, so does 2nt2^nt for every nonnegative integer nn.