Let $\ f: \mathbb Q^2\to\mathbb Q^2$ be defined by $\ f(x,y) = (x-3y , x + 3y- 1)$ Prove the function is 1:1, onto, find the inverse and $\ f \circ f $.

I was able to prove 1:1 by showing $\ f(x_1,y_1)=f(x_2,y_2) \implies (x_1,y_1)=(x_2,y_2) $, which was relatively simple, but i'm having a hard time using the definition of the onto function with this 2 variable function.

Any push in the right direction would be helpful.

  • $\begingroup$ Hint: Given $(r,s)\in\mathbb{Q}^2$ you must show that there exist $(x,y)\in\mathbb{Q}^2$ such that \begin{align*} x-3y&=r\\ x+3y-1&=s \end{align*} $\endgroup$ – Ángel Mario Gallegos Apr 12 '16 at 19:41
  • $\begingroup$ So isolating x and y and showing that they belong to Q is proof enough? $\endgroup$ – Jonathan Apr 12 '16 at 19:54
  • $\begingroup$ Yes, it is enough. $\endgroup$ – Ángel Mario Gallegos Apr 12 '16 at 19:56
  • $\begingroup$ Start by finding the inverse (which is not difficult). $\endgroup$ – almagest Apr 12 '16 at 19:56

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