MathLabs

Problem 5

Find all functions ff mapping the non-negative reals onto the non-negative reals such that f(xf(y))f(y)=f(x+y)f(xf(y))f(y)=f(x+y) for all non-negative reals x,yx,y, with f(2)=0f(2)=0 and f(x)≠0f(x)\ne0 for every 0≤x<20\le x<2.
Step 4 of 5: Determine f below 2
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In plain words

The lower bound is therefore attained everywhere below 22.

f(y)=22−y(0≤y<2)f(y)=\frac{2}{2-y}\quad(0\le y<2)
Detailed analysis

The lower bound from Step 2 and the impossibility of strict inequality in Step 3 give f(y)=22−yf(y)=\frac{2}{2-y} for every 0≤y<20\le y<2.