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(x1​)=x
What is the largest set of real numbers that can be in the domain of f?
Answer Choices:
A. {x∣xî€ =0}
B. {x∣x<0}
C. {x∣x>0}
D. {x∣xî€ =−1 and xî€ =0 and xî€ =1}
E. {−1,1}
Solution:
The conditions on f imply that both
x=f(x)+f(x1​) and x1​=f(x1​)+f(1/x1​)=f(x1​)+f(x)
Thus if x is in the domain of f, then x=1/x, so x=±1.
The conditions are satisfied if and only if f(1)=1/2 and f(−1)=−1/2​.
The problems on this page are the property of the MAA's American Mathematics Competitions