Image of set

**JG89** · October 26th 2010, 06:34 PM

Let $f(x,y) = (x^2 - y^2, 2xy)$ where f is defined on the open set $A = \{ (x,y) \in \mathbb{R}^2 : x > 0 \}$ .

Question: What is the set f(A)?

I'm guessing they're asking me to describe f(A) using set notation. But I don't know how to do this. I've tried plotting a set of points to get an idea what f(A) would look like on the plane, but I can't get any idea whatsoever.

Any help or ideas?

**tonio** · October 27th 2010, 02:25 AM

Originally Posted by JG89

Let $f(x,y) = (x^2 - y^2, 2xy)$ where f is defined on the open set $A = \{ (x,y) \in \mathbb{R}^2 : x > 0 \}$ .

Question: What is the set f(A)?

I'm guessing they're asking me to describe f(A) using set notation. But I don't know how to do this. I've tried plotting a set of points to get an idea what f(A) would look like on the plane, but I can't get any idea whatsoever.

Any help or ideas?

Hint: prove that for any $(a,b)\in\mathbb{R}^2$ there exist $x,y\in\mathbb{R}$ s.t. $2xy=b\,\,and\,\,x^2-y^2=a$ . This solves your problem.

Tonio

**JG89** · October 27th 2010, 06:34 AM

Tonio, this isn't true. Consider the origin (0,0). If there is x, y such that $2xy = 0, x^2 - y^2 = 0$ then we must have $y = 0$ in order for $2xy = 0$ to hold, since x > 0. Thus $x^2 - y^2 = x^2 > 0$ . So f will never map to the origin.

How about this:

Think about any point (x,y) in A as the complex number $x + iy = re^{i \theta}$ where $- \pi / 2 < \theta < \pi / 2$ .Then $f(x+iy) = (x+iy)^2 = (re^{i\theta})^2 = r^2 e^{i2\theta}$ . Let $r' = r^2, 2\theta = \phi$ . Then $r^2 e^{i 2 \theta} = r' e^{i \phi}$ where $- \pi < \phi < \pi$ .

Thus $f(A) = \{(x,y) \in \mathbb{R}^2 : (x,y) \neq (x, 0) for x <= 0 \}$

**tonio** · October 27th 2010, 10:47 AM

Originally Posted by JG89

Tonio, this isn't true. Consider the origin (0,0). If there is x, y such that $2xy = 0, x^2 - y^2 = 0$ then we must have $y = 0$ in order for $2xy = 0$ to hold, since x > 0. Thus $x^2 - y^2 = x^2 > 0$ . So f will never map to the origin.

How about this:

Think about any point (x,y) in A as the complex number $x + iy = re^{i \theta}$ where $- \pi / 2 < \theta < \pi / 2$ .Then $f(x+iy) = (x+iy)^2 = (re^{i\theta})^2 = r^2 e^{i2\theta}$ . Let $r' = r^2, 2\theta = \phi$ . Then $r^2 e^{i 2 \theta} = r' e^{i \phi}$ where $- \pi < \phi < \pi$ .

Thus $f(A) = \{(x,y) \in \mathbb{R}^2 : (x,y) \neq (x, 0) for x <= 0 \}$

Oops! I didn't see the condition $x>0$ ....jeje

Tonio

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