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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![**Exercise 3: Sketch the Domain of the Function**
Consider the function:
\[ f(x, y) = \ln(4x^2 - y) \]
**Objective:** Determine and sketch the domain of the function.
### Explanation of the Function
The given function \( f(x, y) = \ln(4x^2 - y) \) involves a natural logarithm. The domain of the natural logarithm function, \(\ln(u)\), requires that \(u > 0\). Therefore, the expression inside the logarithm, \(4x^2 - y\), must be greater than zero for the function to be defined.
### Domain Condition
\[ 4x^2 - y > 0 \]
Rearranging the inequality gives us:
\[ y < 4x^2 \]
### Graphical Representation
The domain is the set of all points \((x, y)\) such that \(y < 4x^2\). Graphically, this is the region below the parabola defined by the equation \(y = 4x^2\).
### Graph Description
- The parabola \(y = 4x^2\) opens upwards and is symmetric about the y-axis.
- The vertex of the parabola is at the origin \((0, 0)\).
- The domain consists of all the points below this parabola.
Therefore, the domain can be illustrated by shading the region below the parabola on the xy-plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93c425ca-5be8-4a48-b5f2-917fc035445e%2F5132c28a-15e5-463e-a956-6805cdaa624e%2Fyl9yztg_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise 3: Sketch the Domain of the Function**
Consider the function:
\[ f(x, y) = \ln(4x^2 - y) \]
**Objective:** Determine and sketch the domain of the function.
### Explanation of the Function
The given function \( f(x, y) = \ln(4x^2 - y) \) involves a natural logarithm. The domain of the natural logarithm function, \(\ln(u)\), requires that \(u > 0\). Therefore, the expression inside the logarithm, \(4x^2 - y\), must be greater than zero for the function to be defined.
### Domain Condition
\[ 4x^2 - y > 0 \]
Rearranging the inequality gives us:
\[ y < 4x^2 \]
### Graphical Representation
The domain is the set of all points \((x, y)\) such that \(y < 4x^2\). Graphically, this is the region below the parabola defined by the equation \(y = 4x^2\).
### Graph Description
- The parabola \(y = 4x^2\) opens upwards and is symmetric about the y-axis.
- The vertex of the parabola is at the origin \((0, 0)\).
- The domain consists of all the points below this parabola.
Therefore, the domain can be illustrated by shading the region below the parabola on the xy-plane.
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