# Equation for the image of a vector-valued function

$\vec{X}(s,t)=(s^2cost,s^2sint,s), -3\leq s\leq 3, 0\leq t\leq 2\pi$.
So I've already found the normal vector at (s,t)=(-1,0) to be (-1,0,-2) and found the tangent plane at (1,0,-1) to be $x+2z=-1$ (feel free to correct me if either of those end up being wrong and important to my question).
The question reads "Find an equation for the image of $\vec{X}$ in the form $F(x,y,z)=0$.
What I initially did was to say that $x=s^2cost$, $y=s^2sint$, and $z(x,y)=(x^2+y^2)^{\frac{1}{4}}$, which then just gives the equation $F(x,y,z)=(x^2+y^2)^{\frac{1}{4}}-z=0$, but this seems suspiciously simplistic. Is there some other way to define F that doesn't require that z be defined explicitly in terms of x and y, or is what I've written really all that there is to the problem?