The graph of the function $y=x+\frac{1}{x}$ is a hyperbola. We can see this more clearly by writing it out in the standard form $x^{2}-x y+1=0$ or $\left(\frac{y}{2}\right)^{2}-\left(x-\frac{1}{2} y\right)^{2}=1$. The hyperbola has asymptotes given by $x=0$ and $y=x$, so the lines of symmetry will be the (interior and exterior) angle bisectors of these two lines. This means that they will be $y=\tan \left(67.5^{\circ}\right) x$ and $y=-\cot \left(67.5^{\circ}\right) x$, which, using the tangent half-angle formula $\tan \left(\frac{x}{2}\right)=\sqrt{\frac{1+\cos (x)}{1-\cos (x)}}$, gives the two lines $y=(1+\sqrt{2}) x$ and $y=(1-\sqrt{2}) x$.