Find the map of the closed ball $B(0,1)$ of the following continuous function $f(x,y,z)=(\frac x3,\frac y2-1,\frac z9+1)$ and $f^{-1}(0)$.

Question

Find the map of the closed ball $B(0,1)$ of the following continuous function $$f(x,y,z)=\left(\frac x3,\frac y2-1,\frac z9+1\right)$$ and $f^{-1}(0)$.

---

$f^{-1}$ seems quite simple, I got $(0,2,-9)$, now when it comes to this map of the closed ball, I'm at a stop because I'm unsure if i do (let's say X is the map) $X=\{x: d(f(0),f(x))\le |f(1,0,0)|\}$ is incorrect?

Answer 1

Finding the image $f(B)$ is a three step process.

1. Introduce names for new coordinates, say $u=\frac x3$, $v=\frac y2-1$, $w=\frac z9+1$.
2. Express old coordinates in terms of new: $x=3u$ and similarly for others.
3. Plug the formulas from (2) into the inequality $x^2+y^2+z^2\le 1$ that describes $B(0,1)$.

Following these steps, you should get an inequality that describes a solid ellipsoid.