Let $\Phi(u,v)=(u^2,v)$. Is $\Phi$ one-to-one? If not, determine a domain on which $\Phi$ is one-to-one.

Find the image under $\Phi$ of:

$(\mathbf{a})$The $u$- and $v$-axes

A. $0 < x < \infty ,\quad -\infty < y < \infty$
B. $-\infty \le x < \infty ,\quad -\infty < y < \infty$
C. $0 \le x < \infty ,\quad 0 < y < \infty$
D. $0 \le x < \infty ,\quad -\infty < y < \infty$
E. $-\infty < x < \infty ,0 \le y < \infty$

$(\mathbf{b})$The rectangle $\mathcal{R}=[-3,3]\times [-3,3]$

A. $[0,9]\times [-3,3]$
B. $[-9,9]\times [-9,9]$
C. $[0,9]\times [0,3]$
D. $[0,3]\times [-9,9]$
E. $[-3,3]\times [0,9]$

$(\mathbf{c})$The line segment joining $(0,0)$ and $(6,6)$

A. $y=\sqrt{x},\quad 0 \le x \le 6$
B. $y=x,\quad 0 \le x \le 6$
C. $y=\sqrt{x},\quad 0 \le x \le 36$
D. $y=x^2,\quad 0 \le x \le 36$
E. $y=x,\quad 0 \le x \le 36$

$(\mathbf{d})$The triangle with vertices $(0,0) , (0,6)$ and $(6,6)$

A. $0 \le y \le x,\quad 0 \le x \le 36$
B. $0 \le y \le \sqrt{x},\quad 0 \le x \le 6$
C. $0 \le y \le x,\quad 0 \le x \le 6$
D. $0 \le y \le x^2,\quad 0 \le x \le 36$
E. $\sqrt{x} \le y \le 6,\quad 0 \le x \le 36$