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...
Related questions
Concept explainers
Transformation of Graphs
The word ‘transformation’ means modification. Transformation of the graph of a function is a process by which we modify or change the original graph and make a new graph.
Exponential Functions
The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
Describe the domain of the following function
![### Mathematical Function:
The given function is represented as:
\[
(\alpha) \, h(x) = \frac{x - 2}{\sqrt{x^2 - 2}}
\]
### Explanation:
This is a rational function where:
- **Numerator:** \( x - 2 \)
- **Denominator:** \( \sqrt{x^2 - 2} \)
### Key Points to Consider:
1. **Domain:**
- The function is defined where the expression within the square root is non-negative. Therefore, \( x^2 - 2 \geq 0 \).
- Solving \( x^2 - 2 = 0 \), we get \( x = \pm \sqrt{2} \).
- The domain is \( x \leq -\sqrt{2} \) or \( x \geq \sqrt{2} \).
2. **Behavior:**
- The function may have vertical asymptotes or undefined points at \( x = \pm \sqrt{2} \) since the denominator is zero at these points.
- The numerator \( x - 2 \) results in a zero at \( x = 2 \).
Understanding the behavior of this function involves analyzing where it is defined and potential asymptotic behavior as \( x \) approaches certain values.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fbf8bb-16ae-4f14-94e7-14785565b4b4%2F7fa6c122-46c7-4c38-802d-447e01b402f3%2Fgknywpa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Function:
The given function is represented as:
\[
(\alpha) \, h(x) = \frac{x - 2}{\sqrt{x^2 - 2}}
\]
### Explanation:
This is a rational function where:
- **Numerator:** \( x - 2 \)
- **Denominator:** \( \sqrt{x^2 - 2} \)
### Key Points to Consider:
1. **Domain:**
- The function is defined where the expression within the square root is non-negative. Therefore, \( x^2 - 2 \geq 0 \).
- Solving \( x^2 - 2 = 0 \), we get \( x = \pm \sqrt{2} \).
- The domain is \( x \leq -\sqrt{2} \) or \( x \geq \sqrt{2} \).
2. **Behavior:**
- The function may have vertical asymptotes or undefined points at \( x = \pm \sqrt{2} \) since the denominator is zero at these points.
- The numerator \( x - 2 \) results in a zero at \( x = 2 \).
Understanding the behavior of this function involves analyzing where it is defined and potential asymptotic behavior as \( x \) approaches certain values.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning