Mapping D* onto a Triangle: Investigating One-to-One Functions

[der.physika]

Homework Statement

Let D* = [0,1] x [0,1] and define T on D* by T(x*,y*) = (x*y*, x*). Determine the image set D. Is T one-to-one?

Homework Equations

The Attempt at a Solution

Okay... So I know it is not one to one, if you take out the point (x=0) then it is one-to-one, so you must be careful with the origin.
But I don't know how to find the image set D. Because there are two variables in the first component in T x and y*. How do I find D?

 

[tiny-tim]

hi der.physika!

it's easier if you break D* into two functions

what is the image of the function that sends (x, y) to (x, xy)?

(it may help to consider all the (x, 1)s)

 

[der.physika]

hi der.physika!

it's easier if you break D* into two functions

what is the image of the function that sends (x, y) to (x, xy)?

(it may help to consider all the (x, 1)s)

Okay... I think I found the solution

by using parametric equations

(t,0); (1,t); (t,1), (0,t) and plugging them in yields (0,0), (0,0), (t,t), (0,t) which makes it not 1 to 1. So... it gives me the triangle with those vertices. is this correct?

 

[tiny-tim]

(just got up :zzz: …)

… (t,0); (1,t); (t,1), (0,t) and plugging them in yields (0,0), (0,0), (t,t), (0,t) which makes it not 1 to 1. So... it gives me the triangle with those vertices. is this correct?

That's right!

(as you've seen, a short-cut that will work in any "non-folding" case is to simply find where all the corners go to )