MathLabs

Problem 4

Construct a function f:Q+→Q+f:\mathbb Q^+\to\mathbb Q^+ such that f(xf(y))=f(x)/yf(xf(y))=f(x)/y for all positive rational numbers x,yx,y.
Step 5 of 5: Verify the functional equation
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f(xf(y))=f(x)f(f(y))=f(x)yf(xf(y))=f(x)f(f(y))=\frac{f(x)}{y}
Detailed analysis

For this construction, multiplicativity and the prime definition give f(f(y))=1/yf(f(y))=1/y. Hence f(xf(y))=f(x)f(f(y))=f(x)yf(xf(y))=f(x)f(f(y))=\frac{f(x)}{y}, as required.