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.