Chapter 12: Q 50. (page 932)

Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous.
$f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$

Short Answer

Expert verified

The function $f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$ is continuous on the set $\{(x, y, z) \in f^{3} : x^{2} + y^{2} + z^{2} > 0\}$ .

Step by step solution

Step 1. Given information.

We have given expression: $f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$

Determine the domains of the functions.

Consider the function: $g : f^{3} \to f$ .Then the domain of the function $g$ is defined as

$D o m a i n (g) = \{(x, y, z) \in f^{3} : g (x, y, z)\}$

Since the domain of the natural logarithm $\ln (x)$ is the set $\{x \in f : x > 0\}$ .

$l n (x^{2} + y^{2} + z^{2})$ is defined on the set $\{(x, y, z) \in f^{3} : x^{2} + y^{2} + z^{2} > 0\}$ .

Here from the above statement, the domain of the function is role="math" localid="1653402193440" $f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$ is

$D o m a i n (f) = \{(x, y, z) \in f^{3} : x^{2} + y^{2} + z^{2} > 0\}$

Step 3. To find continuous of the function.

Since the natural logarithm is continuous on the entire domain, therefore the function $f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$ is continuous on the set $\{(x, y, z) \in f^{3} : x^{2} + y^{2} + z^{2} > 0\}$

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Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous.
$f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$

Short Answer

Step by step solution

Step 1. Given information.

Determine the domains of the functions.

Step 3. To find continuous of the function.

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Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous.f(x,y,z)=ln(x2+y2+z2)

Short Answer

Step by step solution

Step 1. Given information.

Determine the domains of the functions.

Step 3. To find continuous of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Calculus

Pure Maths

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous.
$f (x, y, z) = \ln (x^{2} + y^{2} + z^{2})$