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 3 of 6: f(1)=1f(1)=1, so ff is an involution
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f(1)=1,f(f(y))=yf(1)=1,\qquad f(f(y))=y
Detailed analysis

Setting y=1y=1 in (i) gives f(xf(1))=f(x)f(xf(1)) = f(x); by injectivity (Step 2), xf(1)=xxf(1)=x for all x>0x>0, so f(1)=1f(1)=1. Plugging back into Step 1, f(f(y))=yf(f(y))=y for all yy.