Need help with level curves

Hello!

Homework Statement

Well,I'm having a problem drawing level curves for piecewise functions.
The problem is, how do I know which value the constant k will hold?

Homework Equations

The functions is the following:

f(x,y)=4 if x^2+y^2<=16
sqrt(32-x^2-y^2) if 16<x^2+y^2<=32

The Attempt at a Solution

The solution I've attempted and which I'm not sure it's correct is:
I've drawn a level curve of level 4,because it's within the domain of f(x,y)(which is ]-infinity;32]) and it's the point where the function changes to the other branch.
Does this make sense?

Just another question,to determine the domain of the second "piece" of the function,why do we also use the sqrt(32-x^2-y^2) condition and not only just the if clause?

Hello!

Homework Statement

Well,I'm having a problem drawing level curves for piecewise functions.
The problem is, how do I know which value the constant k will hold?

Homework Equations

The functions is the following:

f(x,y)=4 if x^2+y^2<=16
sqrt(32-x^2-y^2) if 16<x^2+y^2<=32

The Attempt at a Solution

The solution I've attempted and which I'm not sure it's correct is:
I've drawn a level curve of level 4,because it's within the domain of f(x,y)(which is ]-infinity;32]) and it's the point where the function changes to the other branch.
Does this make sense?

Just another question,to determine the domain of the second "piece" of the function,why do we also use the sqrt(32-x^2-y^2) condition and not only just the if clause?

z = sqrt(32-x^2-y^2) ==> z^2 + x^2+y^2 = (sqrt(32))^2
this is a sphere.

first draw level curves for this,

and then erase all but curves that are between circles with 4 and (sqrt(32))^2

What I don't understand is,why do we use the k constant with value 4 specifically?Why couldn't we use other value?
Also,in the second piece of the function why do we equal sqrt(32-x^2-y^2)=4?Does it have anything to do with the fact that 4 is the point where the function switches to the other branch?

HallsofIvy