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

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.
