Answer
See graph and explanations.
Work Step by Step
Step 1. We are given $y=\frac{x^2+1}{x}=x+\frac{1}{x}$, which can be considered as the sum of two functions.
Step 2. It is easier to graph the function by first plotting the known functions $y=x$ (which is a line) and $f(x)=\frac{1}{x}$ where $f'(x)=-\frac{1}{x^2}\lt0$ and thus decreasing over all its open intervals. The asymptotes are $x=0$ and $y=0$ for this function.
Step 3. Find the first derivative $y'=1-\frac{1}{x^2}$ and $y'=0$ gives $x=\pm1$ which are the critical points (plus when $x=0$). We have the $y'$ sign changes as $..(+)..(-1)..(-)..(0)..(-)..(1)..(+)..$ which gives increasing regions of $(-\infty,-1),(1,\infty)$ and decreasing regions $(-1,0),(0,1)$ and a local maximum at $x=-1$, a local minimum at $x=1$.
Step 4. The new asymptotes are $x=0$ and $y=x$.
Step 5. To graph the given function from $y=x$ and $f(x)=\frac{1}{x}$, use the above information and add the corresponding y-coordinates at some key points and connect the points to form curves as shown in the figure.
