hi!Ronald again.I am a student.I have a mathematics problem.It is about secondary level.
The question is as following:
Let S be he set of real no. strictly greater than -1 .Find all functions f:S --> S statisfying the two conditions:
1)f(x+f(y)+xf(y))=y+f(x)+yf(x) for all x and y in S
2)f(x)/x is strictly increasing on each of the interval -1<x<0 and 0<x.
Could you help me to solve it .Thank you!!!
If you write g(x,y)=x+f(y)+xf(y) then your first condition is
Thus f(f(g(x,y)))=f(g(y,x))=g(x,y) and hence for t in the range of g, f(f(t))=t.
Notice that g(x,y)=f(y)+x(1+f(y)) and f(y) > -1 so for any fixed y and sufficiently large x, g(x,y) > 0. Hence there are positive t's in the range of g.
Condition 2) is that f(x)/x is strictly increasing and thus, for positive x, f(f(x))/f(x) is also increasing. Therefore for positive x, (f(f(x))/f(x))*(f(x)/x) is increasing.
But also, for positive x in the range of g, (f(f(x))/f(x))*(f(x)/x)=(x/f(x))*(f(x)/x)=1 which is impossible. Hence there is no function f that satisfies the conditions 1) and 2).
Saroop and Chris
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