Give the explicit form of the following parametrized surface Let $\boldsymbol{X}:\boldsymbol{R}^2\to \boldsymbol{R}^3$ be the paramtrized surface given by$$\boldsymbol{X}(s,t)=(s^2-t^2,s+t,s^2+3t)$$
I'm trying to describe the parametrized surface by an equation of the form $z=f(x,y)$

You have three equations: $x=s^2-t^2$, $y=s+t$ and $z=s^2+3t$. You need to eliminate $s$ and $t$.
Rearranging $y=s+t$ gives $t=y-s$. Substituting this into the other two equations gives
$$\begin{eqnarray*}
x &=& 2ys-y^2 \\
z&=& s^2-3s+3y
\end{eqnarray*}$$
Now we can rearrange $x = 2ys-y^2$ to solve for $s$. We get $\displaystyle{s=\frac{x+y^2}{2y}}$.
Finally, substituting this into $z= s^2-3s+3y$ gives an equation in $x$, $y$ and $z$:
$$z = \left(\frac{x+y^2}{2y}\right)^{\!2} - 3\left(\frac{x+y^2}{2y}\right) + 3y$$
This assumes that $y \neq 0$. If $y=0$ then $s=-t$, meaning that $x=0$ and $z=t^2+3t$. Here $t$ is arbitrary. Completing the square gives
$$z=\left(t+\frac{3}{2}\right)^{\!2} - \frac{9}{4}$$
This tells is that for al $t$, $z \ge -\frac{9}{4}$. So, when $x=y=0$, we have to add the semi-axis $z \ge -\frac{9}{4}$.
