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Blericker74 2022-07-08 Answered
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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Answers (1)

SweallySnicles3
Answered 2022-07-09 Author has 21 answers
I do not think you define the concept of "even" or "odd" 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.
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