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ravi623

  • 2 years ago

show that there is only one way to write f as the sum of an even function and an odd function

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  1. satellite73
    • 2 years ago
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    one way is \[\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}\] oh only one way...

  2. satellite73
    • 2 years ago
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    usual idea is to imagine there are are two ways and then show they are the same. so maybe put \[f(x)=e(x)+o(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}\] where \(o(x)\) is odd and \(e(x)\) is even and then show that \(e(x)=\frac{f(x)+f(-x)}{2}\) and \(o(x)=\frac{f(x)-f(-x)}{2}\)

  3. satellite73
    • 2 years ago
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    yes, that should work let me know if it is not clear

  4. ravi623
    • 2 years ago
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    nice explanation, thanks!!

  5. satellite73
    • 2 years ago
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    yw

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