### Evaluation of the Function $ g(x, y, z) = \ln(x + y + z) $Below are the given functions to evaluate. If an answer does not exist, enter DNE (Does Not Exist).#### (a) $ g(0, 0, 0) $\[ g(0, 0, 0) = \ln(0 + 0 + 0) = \ln(0) \]Since the natural logarithm of zero is undefined, the answer is: DNE.#### (b) $ g(1, 0, 0) $\[ g(1, 0, 0) = \ln(1 + 0 + 0) = \ln(1) \]Since the natural logarithm of one is zero, the answer is:\[ 0 \]#### (c) $ g(0, 1, 0) $\[ g(0, 1, 0) = \ln(0 + 1 + 0) = \ln(1) \]Since the natural logarithm of one is zero, the answer is:\[ 0 \]#### (d) $ g(z, x, y) $\[ g(z, x, y) = \ln(z + x + y) \]Here, the function's form doesn't change; it is simply restructured.#### (e) $ g(x + h, y + k, z + l) $\[ g(x + h, y + k, z + l) = \ln((x + h) + (y + k) + (z + l)) \]This simplifies to:\[ g(x + h, y + k, z + l) = \ln(x + y + z + h + k + l) \]Here, the inputs are expressed as summing each variable $x, y, z$ with additional parameters $h, k, l$.### Notes- The natural logarithm function $\ln$ is defined only for positive arguments. When evaluating these functions, ensure the sum $x + y + z$ is positive; otherwise, the function does not exist (DNE).- Be mindful of the domain restrictions for the logarithmic function when calculating $ g(x, y, z) $. The function \( \ln(x + y + z

Answered: or the function, evaluate the…

or the function, evaluate the following. (If an answer does not exist, enter DNE.) д(x, у, 2) In(x + y + z) a) g(0, 0, 0) b) g(1, 0, 0) с) g(0, 1, 0) Ed) g(z, х, у) e) д(x + h, у + k, z + )

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Question

### Evaluation of the Function $ g(x, y, z) = \ln(x + y + z) $

Below are the given functions to evaluate. If an answer does not exist, enter DNE (Does Not Exist).

#### (a) $ g(0, 0, 0) $
\[ g(0, 0, 0) = \ln(0 + 0 + 0) = \ln(0) \]
Since the natural logarithm of zero is undefined, the answer is: DNE.

#### (b) $ g(1, 0, 0) $
\[ g(1, 0, 0) = \ln(1 + 0 + 0) = \ln(1) \]
Since the natural logarithm of one is zero, the answer is:
\[ 0 \]

#### (c) $ g(0, 1, 0) $
\[ g(0, 1, 0) = \ln(0 + 1 + 0) = \ln(1) \]
Since the natural logarithm of one is zero, the answer is:
\[ 0 \]

#### (d) $ g(z, x, y) $
\[ g(z, x, y) = \ln(z + x + y) \]
Here, the function's form doesn't change; it is simply restructured.

#### (e) $ g(x + h, y + k, z + l) $
\[ g(x + h, y + k, z + l) = \ln((x + h) + (y + k) + (z + l)) \]
This simplifies to:
\[ g(x + h, y + k, z + l) = \ln(x + y + z + h + k + l) \]
Here, the inputs are expressed as summing each variable $x, y, z$ with additional parameters $h, k, l$.

### Notes
- The natural logarithm function $\ln$ is defined only for positive arguments. When evaluating these functions, ensure the sum $x + y + z$ is positive; otherwise, the function does not exist (DNE).
- Be mindful of the domain restrictions for the logarithmic function when calculating $ g(x, y, z) $. The function \( \ln(x + y + z

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