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
![## Problem 5: Sketch the Graph
**Equation:**
\[ y = 2 - (7)^x \]
**Instructions:**
- Label any asymptotes.
- Identify the y-intercept(s).
- State the domain and range.
### Detailed Explanation
**Asymptotes:**
- Determine if there is a horizontal asymptote, which occurs as \(x\) approaches infinity. Analyze the constant in the equation to identify the asymptote line.
**Y-intercept:**
- Find the y-intercept by setting \(x = 0\). Substitute into the equation to solve for \(y\).
**Domain and Range:**
- **Domain:** All possible values of \(x\). For this equation, consider if there are any restrictions on \(x\).
- **Range:** All possible values of \(y\), which depend on the behavior of the function as \(x\) changes.
Make sure to sketch the curve considering the behavior of an exponential function, where it grows or decays rapidly, and understand the transformations applied in this specific function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7440da40-7bdb-4d11-be8e-b7bdca03efe4%2Fdcca5e38-5f34-4327-b405-1d707f32dda9%2F4dhbcxf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem 5: Sketch the Graph
**Equation:**
\[ y = 2 - (7)^x \]
**Instructions:**
- Label any asymptotes.
- Identify the y-intercept(s).
- State the domain and range.
### Detailed Explanation
**Asymptotes:**
- Determine if there is a horizontal asymptote, which occurs as \(x\) approaches infinity. Analyze the constant in the equation to identify the asymptote line.
**Y-intercept:**
- Find the y-intercept by setting \(x = 0\). Substitute into the equation to solve for \(y\).
**Domain and Range:**
- **Domain:** All possible values of \(x\). For this equation, consider if there are any restrictions on \(x\).
- **Range:** All possible values of \(y\), which depend on the behavior of the function as \(x\) changes.
Make sure to sketch the curve considering the behavior of an exponential function, where it grows or decays rapidly, and understand the transformations applied in this specific function.
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