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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![### Problem 38
#### Task:
Find and sketch the domains of the following functions:
\[ f(x, y) = \frac{e^{-y}}{\sqrt{xy}} \]
\[ g(x, y) = \frac{xy}{\ln(9 - x^2 - 9y^2)} \]
#### Steps to Determine the Domains:
##### For \( f(x, y) = \frac{e^{-y}}{\sqrt{xy}} \):
1. **Identify the restrictions on \(x\) and \(y\)**:
- The denominator \(\sqrt{xy}\) must be positive since you cannot take the square root of a negative number or zero.
- Therefore, \( x > 0 \) and \( y > 0 \).
2. **Express the Conditions**:
- The domain of \( f(x, y) \) is given by the set:
\[
D_f = \{ (x, y) \in \mathbb{R}^2 \mid x > 0, y > 0 \}
\]
##### For \( g(x, y) = \frac{xy}{\ln(9 - x^2 - 9y^2)} \):
1. **Identify the restrictions on \(x\) and \(y\)**:
- The argument of the logarithmic function \(\ln(9 - x^2 - 9y^2)\) must be positive: \( 9 - x^2 - 9y^2 > 0 \).
- Therefore, \( x^2 + 9y^2 < 9 \):
- This inequality describes an ellipse centered at the origin with semi-major axis 3 (along the x-axis) and semi-minor axis 1 (along the y-axis).
- The logarithmic function \(\ln(9 - x^2 - 9y^2)\) must not equal zero, which implies \( 9 - x^2 - 9y^2 \neq 1 \).
2. **Express the Conditions**:
- The domain of \( g(x, y) \) is given by the set:
\[
D_g = \left\{ (x, y) \in \mathbb{R}^2 \mid x^2 + 9y^2 < 9 \text](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18107b7c-8d8e-412b-a2cd-6addf4186be2%2F84183924-3e42-4dca-a686-ea010d3cd71e%2Fmwayng8_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 38
#### Task:
Find and sketch the domains of the following functions:
\[ f(x, y) = \frac{e^{-y}}{\sqrt{xy}} \]
\[ g(x, y) = \frac{xy}{\ln(9 - x^2 - 9y^2)} \]
#### Steps to Determine the Domains:
##### For \( f(x, y) = \frac{e^{-y}}{\sqrt{xy}} \):
1. **Identify the restrictions on \(x\) and \(y\)**:
- The denominator \(\sqrt{xy}\) must be positive since you cannot take the square root of a negative number or zero.
- Therefore, \( x > 0 \) and \( y > 0 \).
2. **Express the Conditions**:
- The domain of \( f(x, y) \) is given by the set:
\[
D_f = \{ (x, y) \in \mathbb{R}^2 \mid x > 0, y > 0 \}
\]
##### For \( g(x, y) = \frac{xy}{\ln(9 - x^2 - 9y^2)} \):
1. **Identify the restrictions on \(x\) and \(y\)**:
- The argument of the logarithmic function \(\ln(9 - x^2 - 9y^2)\) must be positive: \( 9 - x^2 - 9y^2 > 0 \).
- Therefore, \( x^2 + 9y^2 < 9 \):
- This inequality describes an ellipse centered at the origin with semi-major axis 3 (along the x-axis) and semi-minor axis 1 (along the y-axis).
- The logarithmic function \(\ln(9 - x^2 - 9y^2)\) must not equal zero, which implies \( 9 - x^2 - 9y^2 \neq 1 \).
2. **Express the Conditions**:
- The domain of \( g(x, y) \) is given by the set:
\[
D_g = \left\{ (x, y) \in \mathbb{R}^2 \mid x^2 + 9y^2 < 9 \text
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