First slide

Functions (XII)

Question

A function F(x) satisfies the functional equation $x^{2} F (x) + F (1 - x) = 2 x - x^{4}$ for all real x. F(x) must be

Moderate

A

$x^{2}$
B

$1 - x^{2}$
C

$1 + x^{2}$
D

$x^{2} + x + 1$

Solution

$\begin{array}{l} x^{2} F (x) + F (1 - x) = 2 x - x^{4} (1) \\ Replacing x by 1 - x, we get \\ {(1 - x)}^{2} F (1 - x) + F (x) = 2 (1 - x) - {(1 - x)}^{4} (2) \end{array}$
Eliminating Eliminating F(1 -x) from (1) and (2), we get $F (x) = 1 - x^{2}$

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