Problem:
The function f has the property that for each real number x in its domain, 1/x is also in its domain and
f(x)+f(x1β)=x
What is the largest set of real numbers that can be in the domain of f?
Answer Choices:
A. {xβ£xξ =0}
B. {xβ£x<0}
C. {xβ£x>0}
D. {xβ£xξ =β1 and xξ =0 and xξ =1}
E. {β1,1}
Solution:
The conditions on f imply that both
x=f(x)+f(x1β) and x1β=f(x1β)+f(1/x1β)=f(x1β)+f(x)
Thus if x is in the domain of f, then x=1/x, so x=Β±1.
The conditions are satisfied if and only if f(1)=1/2 and f(β1)=β1/2β.
The problems on this page are the property of the MAA's American Mathematics Competitions