Find all functions f : R → R such that x 2 ⋅ f ( x ) + f ( 1 − x ) = 2 x − x 4 ∀ x ∈ R My solution:Replace x by ( 1 − x ) and by eliminating f ( 1 − x )I obtained f ( x ) = 1 − x 2

Question

Find all functions   f  :      R    →      R   such that       x    2    ⋅  f  (  x  )  +  f  (  1  −  x  )  =  2  x  −      x    4     ∀    x  ∈      R  My solution:Replace x by   (  1  −  x  ) and by eliminating   f  (  1  −  x  )I obtained   f  (  x  )  =  1  −      x    2

inmholtau5 · Accepted Answer

Your solution is incomplete, because the solution is not required to be continuous. Your system of equations      (                                        x            2                                    1                                      1                          (          1          −          x                      )            2                                )        (                            f          (          x          )                                      f          (          1          −          x          )                      )    =      (                            2          x          −                      x            4                                                2          (          1          −          x          )          −          (          1          −          x                      )            4                                )  has infinitely many solutions if   x  =  φ  =            1      +              5              2   or   x  =  1  −  φ  =            1      −              5              2  . In this case, the matrix is not invertible, and we get      (                            f          (          φ          )                                      f          (          1          −          φ          )                      )    =      (                            −          φ                                      φ          −          1                      )    +  λ      (                            −          1                                      1          +          φ                      )  So for each   λ  ∈      R  , the following is a valid solution:  f  (  x  )  =      {                            1          −                      x            2                                     for           x          ∈                      R                    ∖          {          φ          ,          1          −          φ          }                                      −          φ          −          λ                           for           x          =          φ                                      φ          −          1          +          λ          (          1          +          φ          )                           for           x          =          1          −          φ

Find all functions f:RR -> RR such that x^2 * f(x)+f(1−x)=2x−x^4 AA x in RR

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