If the domain of F (x) is symmetric about the origin, then 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

If the domain of F (x) is symmetric about the origin, then F1 (x) = f (x) + F (- x) is an even function and F2 (x) = f (x) - f (- x) is an odd function

F1 (- x) = f (- x) + F (- x)) = f (x) + F (- x) = F1 (x). Because the domain of F (x) is symmetric about the origin, F1 (x) is an even function
F 2 (- x) = f (- x) - f (- x)) = f (- x) - f (x) = - F 2 (x). Because the domain of F (x) is symmetric about the origin, F 2 (x) is an odd function
The condition of even function is: F (x) = f (- x), and its domain is symmetric about the origin
The condition of odd function is: F (x) = - f (- x), and its domain is symmetric about the origin

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