Describe in words the region in space where the function f x

Describe (in words) the region in space where the function f (x, y, z) = ln (x^2 + y^2 + z^2 - 2) + Squareroot 9 - x^2 - y^2 - z^2 is continuous.

Solution

Function f is consist of logrithmic function and square root function.

=> Log can not be non-positive and Square root can not be negative.

And also we that x² + y² + z² = r² is the equation of sphere with radius r and center at (0,0,0).

Since log function can not be non-positive, therefore x² + y² + z² > 2.

Also square root can not be negative, therefore 9 - x² + y² + z² >= 0.

=> x² + y² + z² <= 3²

So f is defined such that 2 < x² + y² + z² <= 9.

=> f is defined in the region between the spheres of radius 2^1/2 and 3 with center at (0,0,0).