Since every function can be divided in a even and a odd part: f e ( x ) = f ( x ) + f ( − x ) 2 f o ( x ) = f ( x ) − f ( − x ) 2 f ( x ) = f e ( x ) + f o ( x )How can I obtain the function by it's even part?

Question

Since every function can be divided in a even and a odd part:      f    e    (  x  )  =            f      (      x      )      +      f      (      −      x      )        2        f    o    (  x  )  =            f      (      x      )      −      f      (      −      x      )        2    f  (  x  )  =      f    e    (  x  )  +      f    o    (  x  )How can I obtain the function by it&#039;s even part?

latatuy · Accepted Answer

You can&#039;t. Just consider for example even functions : they have the function 0 as odd part so you can&#039;t recover the function from the function 0.

Krish Logan · Accepted Answer

You can&#039;t obtain f from just fe or just fo. Information gets lost. For example take a function f and then define   g  (  x  )  =  f  (  x  )  +  sin  ⁡  (  x  ) and   h  (  x  )  =  f  (  x  )  +  cos  ⁡  (  x  ), then we have       g          e        (  x  )  =      f          e        (  x  ) and       g          o        (  x  )  =      f          o        (  x  )  +  sin  ⁡  (  x  ) as well as       h          e        (  x  )  =      f          e        (  x  )  +  cos  ⁡  (  x  ) and       h          o        (  x  )  =      f          o        (  x  ). As you can see the even parts can be identical while the odd parts aren&#039;t and vice versa

Since every function can be divided in a even and a odd part: f_e(x)=(f(x)+f(−x))/(2) f_o(x)=(f(x)−f(−x))/(2) f(x)=f_e(x)+f_o(x). How can I obtain the function by it's even part?

Answered question

Answer & Explanation

New Questions in Algebra I