# Linear transformations with functions

1. May 2, 2010

### Xyius

For the linear transformation,

$$T: R^2\rightarrow R^2, T(x,y) = (x^2,y)$$

find the preimage of..
$$f(x)= 2x+1$$

I have no trouble with these types of problems when it comes to vectors that aren't functions. Any help would be appreciated! Thanks!

~Matt

2. May 3, 2010

So you are looking for (x, y) such that $T(x, y)= (x^2, y)= (a, 2a+1)$ for some number a. Okay, since $x^2= a$, x can be either $\sqrt{a}$ or $-\sqrt{a}$. And, of course, y= 2a+ 1. That is, the preimage is the set in $R^2$ $\{(x,y)| y= 2\sqrt{x}+ 1 or y= -2\sqrt{x}+ 1\}$.