Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Understanding Exponential Functions: Long Run Behavior
In this section, we will analyze the long-run behavior of the function:
\[ f(x) = -2 \cdot (4)^x - 3 \]
To understand the behavior of \( f(x) \) as \( x \) approaches positive and negative infinity, consider the questions and explanations provided below.
#### Exploration of \( f(x) \) as \( x \) Approaches Negative Infinity
**Question:**
As \( x \rightarrow -\infty \), \( f(x) \rightarrow \) ?
Options for selection:
- ?
- \(\infty\)
- \(-\infty\)
- 0
- -3
**Explanation:**
Trace the behavior of the function as \( x \) moves toward negative infinity.
#### Exploration of \( f(x) \) as \( x \) Approaches Positive Infinity
**Question:**
As \( x \rightarrow \infty \), \( f(x) \rightarrow \) ?
Options for selection:
- ?
- \(\infty\)
- \(-\infty\)
- 0
- -3
**Explanation:**
Assess the behavior of the function when \( x \) progresses toward positive infinity.
Click the "Next Question" button to continue with the practice and consolidate your learning.
---
**Diagram/Graph Explanation:**
Currently, there is no visual graph provided, but envision the graph of the given exponential function \( f(x) \).
1. **As \( x \) approaches \(-\infty\)**:
- \( 4^x \) becomes a very small positive number approaching 0 because any positive number raised to a large negative power tends to zero.
- Therefore, \( f(x) \approx -3 \).
2. **As \( x \) approaches \(\infty\)**:
- \( 4^x \) grows exponentially. Since it's multiplied by \(-2\), the term \(-2 \cdot 4^x\) will become very large but negative.
- Thus, \( f(x) \rightarrow -\infty \).
Understanding these behaviors will help you in analyzing the general trends of exponential functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ed96b38-bb6c-4f0b-a794-43dac03e0984%2Fd564f1d4-4f5e-4b39-8064-af24338a0087%2Fhp9zdfj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Exponential Functions: Long Run Behavior
In this section, we will analyze the long-run behavior of the function:
\[ f(x) = -2 \cdot (4)^x - 3 \]
To understand the behavior of \( f(x) \) as \( x \) approaches positive and negative infinity, consider the questions and explanations provided below.
#### Exploration of \( f(x) \) as \( x \) Approaches Negative Infinity
**Question:**
As \( x \rightarrow -\infty \), \( f(x) \rightarrow \) ?
Options for selection:
- ?
- \(\infty\)
- \(-\infty\)
- 0
- -3
**Explanation:**
Trace the behavior of the function as \( x \) moves toward negative infinity.
#### Exploration of \( f(x) \) as \( x \) Approaches Positive Infinity
**Question:**
As \( x \rightarrow \infty \), \( f(x) \rightarrow \) ?
Options for selection:
- ?
- \(\infty\)
- \(-\infty\)
- 0
- -3
**Explanation:**
Assess the behavior of the function when \( x \) progresses toward positive infinity.
Click the "Next Question" button to continue with the practice and consolidate your learning.
---
**Diagram/Graph Explanation:**
Currently, there is no visual graph provided, but envision the graph of the given exponential function \( f(x) \).
1. **As \( x \) approaches \(-\infty\)**:
- \( 4^x \) becomes a very small positive number approaching 0 because any positive number raised to a large negative power tends to zero.
- Therefore, \( f(x) \approx -3 \).
2. **As \( x \) approaches \(\infty\)**:
- \( 4^x \) grows exponentially. Since it's multiplied by \(-2\), the term \(-2 \cdot 4^x\) will become very large but negative.
- Thus, \( f(x) \rightarrow -\infty \).
Understanding these behaviors will help you in analyzing the general trends of exponential functions.
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