MathLabs

Problem 1

Find all functions ff defined on the set of positive real numbers which take positive real values and satisfy the two conditions: (i) f(xf(y))=yf(x)f(xf(y)) = yf(x) for all positive real numbers x,yx, y; (ii) f(x)→0f(x) \to 0 as x→∞x \to \infty.
Step 2 of 6: ff is injective
Previous stepNext step
f(a)=f(b)⇒a f(1)=b f(1)⇒a=bf(a)=f(b) \Rightarrow a\,f(1) = b\,f(1) \Rightarrow a=b
Detailed analysis

If f(a)=f(b)f(a)=f(b) then applying ff once more and using Step 1, af(1)=f(f(a))=f(f(b))=bf(1)a f(1) = f(f(a)) = f(f(b)) = b f(1); since f(1)>0f(1)>0, this forces a=ba=b. Hence ff is injective.