If $ \ f: D=[-\pi, \pi] \times [-\pi,\pi] \to \mathbb{R}^3 \ $ , then
What is the range of $ \ f(x,y)=\left( (\cos x+2) \cos y, \ (\cos x+2) \sin y , \ \sin x \right) \ $ ?
Answer:
$ -1 \leq \cos x \leq 1 \\ \Rightarrow 1 \leq \cos x+2 \leq 3 \\ \Rightarrow -1 \leq (\cos x+2) \cos y \leq 3 \ \ on \ \ D \ $
Similarly,
$ 0 \leq (\cos x+2) \sin y \leq 3 \ \ on \ \ D \ $
and
$ 0 \leq \sin x \leq 1 \ \ on \ \ D \ $
Thus the range of $ \ f(x,y) \ $ is the set $ \ S=\{(x,y,z): -1 \leq x \leq 3 , \ 0 \leq y \leq 3 , \ 0 \leq z \leq 1 \} \ $
Am I right ?
Help me out
It should be
$$ \ S=\{(x,y,z): -3 \leq x \leq 3 , \ -3 \leq y \leq 3 , \ -1 \leq z \leq 1 \} \ $$