Problem:
The function f is not defined for x=0, but, for all non-zero real numbers x, f(x)+2f(x1​)=3x. The equation f(x)=f(−x) is satisfied by
Answer Choices:
A. exactly 1 real number
B. exactly 2 real numbers
C. no real numbers
D. infinitely many, but not all, non-zero real numbers
E. all non-zero real numbers
Solution:
Replacing x by x1​ in the given equation, f(x)+2f(x1​)=3x, yields
f(x1​)+2f(x)=x3​.
Eliminating f(x1​) from the two equations yields
f(x)=x2−x2​.
Then f(x)=f(−x) if and only if
x2−x2​=−x2−(−x)2​,
or x2=2. Thus x=±2​ are the only solutions.