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f+ is the even part of the function and f- is the odd part. I'm not able to understand how it is that they got the values of modulus of x and x for the even and odd parts of the function respectively.

For the even part, I think it might be because f(x) would be reflected in the y-axis but then for the even part shouldn't we have modulus of 2x instead of modulus of x?

For the odd part I have no idea how they got it to be x.

Any help would be much appreciated.

user134785
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1 Answers1

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For a general function $f$, you can find the odd and even parts as follows:

If $f(x) = f^+(x) + f^-(x)$, then $f(-x) = f^+(x) - f^-(x)$ (by properties of even/oddness).

Then

$$ \frac12(f(x) + f(-x)) = f^+(x) $$

and

$$ \frac12(f(x) - f(-x)) = f^-(x). $$

BaronVT
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