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 2 of 7: Determine the fixed value
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f(1)=0f(1)=0
Detailed analysis

Putting x=1,y=f(1)x=1,y=f(1) gives f(1)2=f(1)f(1)^2=f(1). If f(1)=1f(1)=1, the original equation with y=1y=1 gives f(x)=xf(x)=x, contradicting boundedness above. Hence f(1)=0f(1)=0.