TITLE: If $f(x)+2f(1/x)=3x$, find all $y$ such that $f(y)=f(-y)$.
QUESTION [5 upvotes]: The function $f(x)$ is not defined when $x=0$. This function has the property that $f(x) + 2f\left(\frac 1x\right) = 3x$. Find all such values of $y$ such that $f(y) = f(-y)$. (This means it is an even function!).

REPLY [12 votes]: First of all, we can find $f$ explicitly. Indeed, we have
$$\begin{align} f(x)+2f(1/x) &= 3x \\ f(1/x)+2f(x)&=3/x\end{align}$$
which is a simple system of linear equations whose solution gives
$$f(x)=\frac{2}{x}-x.$$
Hence, $f(y)=f(-y)$ if and only if $y=\pm \sqrt{2}$.