Use transformation to sketch the graph of the function g, then state the domain, range and horizontal asymptote of the graph. Please show the work and how to complete the problem.
Use transformation to sketch the graph of the function g, then state the domain, range and horizontal asymptote of the graph. Please show the work and how to complete the problem.
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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Use transformation to sketch the graph of the function g, then state the domain, range and horizontal asymptote of the graph. Please show the work and how to complete the problem.
![The given mathematical expression is:
\[ g(x) = 2^{x-4} \]
This function represents an exponential function where the base is 2, and the exponent is \(x - 4\). This implies a horizontal shift of the basic exponential function \(2^x\) to the right by 4 units. There are no accompanying graphs or diagrams with this expression.
Key Characteristics:
- **Domain**: All real numbers.
- **Range**: All positive real numbers.
- **Y-intercept**: At \(x = 4\), \(g(x) = 1\).
- **Asymptote**: Horizontal asymptote at \(y = 0\).
- **Behavior**: The function grows exponentially as \(x\) increases, and approaches zero as \(x\) decreases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcea6d79-c0c2-49cd-aa31-c3eca199518f%2Ffb80ee07-c8cd-4a87-be63-77a607823d1e%2Ftxn5un8_processed.png&w=3840&q=75)
Transcribed Image Text:The given mathematical expression is:
\[ g(x) = 2^{x-4} \]
This function represents an exponential function where the base is 2, and the exponent is \(x - 4\). This implies a horizontal shift of the basic exponential function \(2^x\) to the right by 4 units. There are no accompanying graphs or diagrams with this expression.
Key Characteristics:
- **Domain**: All real numbers.
- **Range**: All positive real numbers.
- **Y-intercept**: At \(x = 4\), \(g(x) = 1\).
- **Asymptote**: Horizontal asymptote at \(y = 0\).
- **Behavior**: The function grows exponentially as \(x\) increases, and approaches zero as \(x\) decreases.
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