A function satisfies the identity $f(x) + 2f\left(\frac1x\right) = 2x+1$ ... find another identity that $f(x)$ satisfies. 
A function satisfies the identity $f(x) + 2f\left( \frac{1}{x} \right) = 2x+1$.
  By replacing all instances of $x$ with $\frac{1}{x}$, find another identity that $f(x)$ satisfies.

I have absolutely no idea what this question is asking, and how to go about it. I would really appreciate some help; thanks in advance!

You have your original identity and new one obtained by substituting $\frac1x$ instead of $x$:
\begin{align}
f(x)+2f\left(\frac1x\right) &= 2x+1 \tag{1}\\
f\left(\frac1x\right)+2f(x) &= \frac2x+1 \tag{2}
\end{align}
Now you want to combine the two equalities to get $f(x)$. If you add to equation $(1)$ the equation $(2)$ multiplied by $-2$, then the occurrences of $f(1/x)$ cancel each other out:
\begin{align*}
-3f(x) &= 2x-\frac4x-1\\
f(x) &= -\frac23x +\frac4{3x}+\frac13
\end{align*}
