[Math] Express $\;f(x)=\frac{x − 1}{x + 1}\;$ as the sum of an even and an odd function.

algebra-precalculus

In homework there is such problem:

Express $\;f(x)=\dfrac{x − 1}{x + 1}\;$ as the sum of an even and an odd function.
(Simplify as much as possible.)

This function is not even and neither odd. Also if we take it as division of 2 functions, neither $x – 1$ nor $x + 1$ are odd or even… so I'm confused…

Best Answer

Define

  • $f_e(x) = \frac12\left(f(x)+f(-x)\right)$
  • $f_o(x) = \frac12\left(f(x)-f(-x)\right)$

Then $f_e$ is even and $f_o$ is odd and $f_e+f_o=f$ for any given $f$

In your special case $\displaystyle f(x) = \frac{x-1}{x+1}$, so $$f_e(x) = \frac12\left(\frac{x-1}{x+1} + \frac{-x-1}{-x+1} \right) = \ldots$$ $$f_o(x) = \frac12\left(\frac{x-1}{x+1} - \frac{-x-1}{-x+1} \right) = \ldots$$ you just have to simplify.

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