How to prove that F1 (x) = f (x) + F (- x) is an even function and F2 (x) = f (x) - f (- x) is an odd function? | Math enthusiast | Mathematical art

How to prove that F1 (x) = f (x) + F (- x) is an even function and F2 (x) = f (x) - f (- x) is an odd function?

F 1 (x) = f (x) + F (- x) f 1 (- x) = f (- x) + F [- (- x)] = f (- x) + F (x) = f (x) + F (- x) = f 1 (x) that is to prove that F 1 (x) is an even function f 2 (x) = f (x) - f (- x) f 2 (- x) = f (- x) - f [- (- x)] = f (- x) - f (x) = - {f (x) - f (- x)} = - F 2 (x) that is to prove that F 2 (x) is an odd function

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