How can we express $f(x) = \ln x, x>0$ as a sum of even and odd function?
We know that every function can be written as a sum of even and odd function. What about this one here? Somebody help.
[Math] Even + Odd Function
calculusfunctionsreal-analysis
Best Answer
Since $\ln x$ is ony defined for $x>0$, you can simply use the even and the odd extension to $\mathbb R$: \begin{align} f_e(x) &= \begin{cases} \tfrac12 ln |x| & x\ne 0\\ 0 & x=0 \end{cases}\\ f_o(x) &= \begin{cases} f_e(x) & x\ge 0\\ -f_e(x) & x<0 \end{cases} \end{align} Obviously $f_e$ is even and $f_o$ is odd. And for $x>0$, $f_e(x)+f_o(x) = \ln x$.