Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = \sin ^ { - 1 } \left( x ^ { 2 } + y ^ { 2 } \right)$$
A) Domain: all points in the xy-plane; range: real numbers $- \frac { \pi } { 2 } \leq z \leq \frac { \pi } { 2 }$; level curves: circles with centers at $( 0,0 )$
B) Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } \leq 1$; range: real numbers $- \frac { \pi } { 2 } \leq z \leq \frac { \pi } { 2 }$; level curves: circles with centers at $( 0,0 )$
C) Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } \leq 1$; range: real numbers $- \frac { \pi } { 2 } \leq z \leq \frac { \pi } { 2 }$; level curves: circles with centers at $( 0,0 )$ and radii $r , 0

Question

Find the domain and range and describe the level curves for the function f(x,y).
-$$f ( x , y ) = \sin ^ { - 1 } \left( x ^ { 2 } + y ^ { 2 } \right)$$
A) Domain: all points in the xy-plane; range: real numbers $- \frac { \pi } { 2 } \leq z \leq \frac { \pi } { 2 }$; level curves: circles with centers at $( 0,0 )$
B) Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } \leq 1$; range: real numbers $- \frac { \pi } { 2 } \leq z \leq \frac { \pi } { 2 }$; level curves: circles with centers at $( 0,0 )$
C) Domain: all points in the $x y$-plane satisfying $x ^ { 2 } + y ^ { 2 } \leq 1$; range: real numbers $- \frac { \pi } { 2 } \leq z \leq \frac { \pi } { 2 }$; level curves: circles with centers at $( 0,0 )$ and radii $r , 0 < r \leq 1$
D) Domain: all points in the $x y$-plane; range: all real numbers; level curves: circles with centers at ( 0 , 0 )

Quizplus · Accepted Answer

The function is defined for all points in the xy-plane, but the range is limited to real numbers between $- \frac{\pi}{2}$ and $\frac{\pi}{2}$. Since $\sin^{-1}(x)$ is only defined for values between -1 and 1, it follows that $x^2 + y^2$ must be less than or equal to 1. Therefore, the domain is all points in the xy-plane satisfying $x^2 + y^2 \leq 1$.

The level curves of the function are the set of points $(x,y)$ in the xy-plane that satisfy $f(x,y) = c$ for some fixed constant $c$. In this case, we have:

$$\sin^{-1}(x^2 + y^2) = c$$

Taking the sine of both sides and using the identity $\sin(\sin^{-1}(x)) = x$, we get:

$$x^2 + y^2 = \sin(c)$$

This equation represents a circle with center at $(0,0)$ and radius $\sqrt{\sin(c)}$. Since the range of the function is limited to values between $- \frac{\pi}{2}$ and $\frac{\pi}{2}$, the radius of the circles is limited to values between 0 and 1. Therefore, the level curves are circles with centers at $(0,0)$ and radii $r$, where $0 < r \leq 1$.

Thus, the correct choice is C.

Find the Domain and Range and Describe the Level Curves f(x,y)=sin⁡−1(x2+y2)f ( x , y ) = \sin ^ { - 1 } \left( x ^ { 2 } + y ^ { 2 } \right)f(x,y)=sin−1(x2+y2)

Find the Domain and Range and Describe the Level Curves $f ( x , y ) = \sin ^ { - 1 } \left( x ^ { 2 } + y ^ { 2 } \right)$