Let f(x, y, z) = √ x + √y + √ z + ln (4 − x 2 − y 2 − z 2 )

Question

Let f(x, y, z) = √x + √y + √z + ln (4 − x² − y² − z² ). Find the domain of f.

Answer 1

Hey Danny,

Whenever a problem asks about the domain, the goal is to look for values (x,y,z) that we cannot use in the function.

For example sqrt(x), we cannot use values of x that are negative.

Because of this we can say that domain for this case alone is x => 0 (greater than or equal to)

Applying this logic to f(x,y,z)=sqrt (x) + sqrt (y) + sqrt (z) we get that (x,y,z =>(0,0,0)

Lastly, we look at the ln (4-x^2-y^2-z^2). For logarithmic functions in general the domain is (0,infinity)

which means that the part enclose in parenthesis has to be greater than 0.

in other words (4-x^2-y^2-z^2) > 0

looking at this we notice that |x|<2,|y|<2 and |z|<2.

when we put both parts together we get that the domain is:

{ (x,y,z) | 0=< X <2 , 0=< Y < 2, 0= < Z < 2}

Let me know if this helps