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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Question
Please find and sketch the domain if the function in the image.
![### Mathematical Function
The function \( f(x, y) \) is defined as follows:
\[ f(x, y) = \ln(x^2 - 25) + \sqrt{-y^2 - x^2 - 10x} \]
#### Detailed Breakdown:
1. **Natural Logarithm Component:**
- The term \( \ln(x^2 - 25) \) represents the natural logarithm of the expression \( x^2 - 25 \).
2. **Square Root Component:**
- The term \( \sqrt{-y^2 - x^2 - 10x} \) represents the square root of the expression \( -y^2 - x^2 - 10x \).
#### Notes:
- The domain of the function includes all values of \( x \) and \( y \) for which both the natural logarithm and the square root components are defined.
- The natural logarithm \( \ln(x^2 - 25) \) requires that \( x^2 - 25 \) be positive, implying \( x > 5 \) or \( x < -5 \).
- The square root \( \sqrt{-y^2 - x^2 - 10x} \) requires that the expression inside it be non-negative, meaning \( -y^2 - x^2 - 10x \geq 0 \).
This function can be used to explore advanced mathematical concepts in multivariable calculus, including domain restrictions and composite functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce103448-bb42-42ed-bd7d-674364c62c4d%2Fd5194e86-361e-43bb-8ddf-df16a55311b8%2Fspaukaa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Function
The function \( f(x, y) \) is defined as follows:
\[ f(x, y) = \ln(x^2 - 25) + \sqrt{-y^2 - x^2 - 10x} \]
#### Detailed Breakdown:
1. **Natural Logarithm Component:**
- The term \( \ln(x^2 - 25) \) represents the natural logarithm of the expression \( x^2 - 25 \).
2. **Square Root Component:**
- The term \( \sqrt{-y^2 - x^2 - 10x} \) represents the square root of the expression \( -y^2 - x^2 - 10x \).
#### Notes:
- The domain of the function includes all values of \( x \) and \( y \) for which both the natural logarithm and the square root components are defined.
- The natural logarithm \( \ln(x^2 - 25) \) requires that \( x^2 - 25 \) be positive, implying \( x > 5 \) or \( x < -5 \).
- The square root \( \sqrt{-y^2 - x^2 - 10x} \) requires that the expression inside it be non-negative, meaning \( -y^2 - x^2 - 10x \geq 0 \).
This function can be used to explore advanced mathematical concepts in multivariable calculus, including domain restrictions and composite functions.
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