<s>[INST] 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... 
 [/INST] I know it's late, but you could also simply multiply both numerator and denominator by $(x-1)$.
$$ \begin{align} 
  \frac{x-1}{x+1} &= \frac{x-1}{x+1} \cdot \frac{x-1}{x-1} \\
  &= \frac{x^2-2x+1}{x^2-1} \\
  &= \frac{x^2+1}{x^2-1} + \frac{2x}{1-x^2} \\
\end{align} $$
The left-hand term is even since it is of the form $even/even$, and the right-hand term is odd since it has the form $odd/even$.
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