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 2 of 5: Prove multiplicativity and surjectivity
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f(xy)=f(x)f(y)f(xy)=f(x)f(y)
Detailed analysis

For any y, use 1=f(1)=f((1/f(y))f(y))=f(1/f(y))/y1=f(1)=f((1/f(y))f(y))=f(1/f(y))/y to get f(1/f(y))=yf(1/f(y))=y. Thus f is onto. Given y choose z=1/f(y), so f(z)=y; then f(xy)=f(xf(z))=f(x)/z=f(x)f(y)f(xy)=f(xf(z))=f(x)/z=f(x)f(y).