Question
Show that if $f$ is any function, then the function $O$ defined by$$O(x)=\frac{f(x)-f(-x)}{2}$$is odd.
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Show that if a function $f$ is defined on an interval symmetric about the origin (so that $f$ is defined at $-x$ whenever it is defined at $x ),$ then $f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}$ (1) Then show that $(f(x)+f(-x)) / 2$ is even and that $(f(x)-f(-x)) / 2$ is odd.
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PROOF Prove that if $f$ is a one-to-one odd function, then $f^{-1}$ is an odd function.
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Show that if a function $f$ is defined on an interval symmetric about the origin (so that $f$ is defined at $-x$ whenever it is defined at $x$ ), then $$ f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2} $$ Then show that $(f(x)+f(-x)) / 2$ is even and that $(f(x)-f(-x)) / 2$ is odd.
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