Relations Question 12
Question: A function F(x) satisfies the functional equation $ x^{2}F(x)+F(1-x)=2x-x^{4} $ for all real x. F(x) must be
Options:
A) $ x^{2} $
B) $ 1-x^{2} $
C) $ 1+x^{2} $
D) $ x^{2}+x+1 $
Show Answer
Answer:
Correct Answer: C
Solution:
$ x^{2}F(x)+F(1-x)=2x-x^{4} $ …(1)
Replacing $ x $ by $ 1-x, $ we get $ {{(1-x)}^{2}}F(1-x)+F(x)=2(1-x)-{{(1-x)}^{4}} $ …(2)
Eliminating $ F(1-x) $ from (1) and (2), we get $ F(x)=1-x^{2}. $
BETA
