Let f be any function which is defined for all numbers. Show that g ( x ) = f ( x ) + f ( − x ) is even. (1) e . g . ⁡ f ( x ) = x 2 ⟹ g ( x ) = x 2 + ( − x ) 2 = 2 x 2 ← even (2) e . g . ⁡ f ( x ) = x 3 ⟹ g ( x ) = x 3 + ( − x ) 3 = 0 ← even (3) e . g . ⁡ f ( x ) = x 3 + x 2 ← neither even nor odd (4) ⟹ g ( x ) = ( x 3 + x 2 ) + ( − x 3 + x 2 ) = 2 x 2 ← even But how it can be proven that the claim holds?And needless to say, can I completely assume "for all numbers" in the problem statements belong to a set of complex numbers(handling imaginary numbers is also required in this problem)?

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Let      f    be any function which is defined for all numbers. Show that      g  (  x  )  =  f  (  x  )  +  f  (  −  x  )    is even.                    (1)                              e          .          g          .                ⁡                 f        (        x        )        =                  x          2                        ⟹                g        (        x        )        =                  x          2                +        (        −        x                  )          2                =        2                  x          2                ←                          even                                (2)                              e          .          g          .                ⁡                 f        (        x        )        =                  x          3                        ⟹                g        (        x        )        =                  x          3                +        (        −        x                  )          3                =        0        ←                          even                                (3)                              e          .          g          .                ⁡                 f        (        x        )        =                  x          3                +                  x          2                                  ←                          neither even nor odd                                (4)                            ⟹                         g        (        x        )        =        (                  x          3                +                  x          2                )        +        (        −                  x          3                +                  x          2                )        =        2                  x          2                ←        even            But how it can be proven that the claim holds?And needless to say, can I completely assume &quot;for all numbers&quot; in the problem statements belong to a set of complex numbers(handling imaginary numbers is also required in this problem)?

SweallySnicles3 · Accepted Answer

I do not think you define the concept of &quot;even&quot; or &quot;odd&quot; functions for functions of a complex variable. I will assume it is real. You have  g  (  −  x  )  =  f  (  −  x  )  +  f  (  −  (  −  x  )  )  =  f  (  −  x  )  +  f  (  x  )  =  g  (  x  )so   g is even.

Let <mtext> </mtext> f <mtext> </mtext> be any function which is defined for a

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