Question: On the set $N\times N$ define $(m,n)\simeq(k,l)$ if $m+l=n+k$.

Draw a sketch of $N\times N$ that shows several equivalence classes. (hint: sketch points on graph paper).

I'm not quite sure how to sketch the equivalence classes. I know that the following is in the relation, but do I just graph the points? Also, how do I find out how many equivalence classes there are and how do I distinguish them? Are there $4$ different equivalence classes ($1$ for $m$, $1$ for $n$, $1$ for $k$, and $1$ for $l$)? I know that if my relation were $\mod p$, there would be $p$ equivalence classes.

Answer: 
What do you know about monotonous functions? Is this function (strictly) monotonous, and if yes, does this tell you something about injectivity?
Hint:

To find the inverse let $y= f^{-1}(x)$ then since $f(f^{-1}(x)) = x$ we have

$$f(f^{-1}(x)) = x = y\sqrt{2+y^2}$$

which gives

$$x^2 = y^2(2+y^2)\to y^4 + 2y^2 - x^2 = 0$$

This is a quadratic equation in $y^2$: $(y^2)^2 + 2y^2 - x^2 = 0$ for which you can use the standard formula on. To determine which root to pick notice that $f^{-1}(0)=0$ and $\text{sign}(f^{-1}(x)) = \text{sign}(x)$.