How to tell if $T(x,y,z) = (y \sin x, z \cos y, xy)$ is one-to-one and/or onto?

Question

$T(x,y,z) = (y \sin x, z \cos y, xy)$ from $\mathbb{R^3} \rightarrow \mathbb{R^3}$

To show 1-to-1, we want to show:

$$y \sin x = y' \sin x' \\ z \cos y = z' \cos y' \\ xy = x'y'$$

I'm not sure what to do here algebraically, since we can't divide anything since we don't know if $x, y, z, x', y', z'$ might be a $0$ value.

For onto, we want to solve for $x, y,$ and $z$:

$$y \sin x = a \\ z \cos y = b \\ xy = c $$

Again, we don't know if any of these values may be a $0$, so we can't be too careful in dividing by $0$. What should I do from here?

Take $x=0=z$ and let $y$ vary. – Git Gud May 18 '14 at 19:58 — Git Gud, May 18 '14 at 19:58
Velcome to the site! – kjetil b halvorsen May 18 '14 at 20:16 — kjetil b halvorsen, May 18 '14 at 20:16

user2566092 · Answer 1 · 2014-05-18T20:04:33.063

4

Hint: For 1-1, consider what you get when $x = 0$ and $z = 0$, and $y$ is any number. For onto, consider what happens to the first coordinate if you set the third coordinate to zero (i.e. $xy = 0$ so either $x = 0$ or $y = 0$).

edited May 18 '14 at 20:04

answered May 18 '14 at 19:59

user2566092

26,142

How to tell if $T(x,y,z) = (y \sin x, z \cos y, xy)$ is one-to-one and/or onto?

1 Answers1