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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![### Question
Which statement describes all the horizontal asymptotes of the function?
\[ f(x) = 2^x + 5 - 3 \]
- [ ] y = 0
- [ ] x = -3
- [ ] x = 0
- [ ] y = -3
**Explanation**
To identify the horizontal asymptotes of the function \( f(x) = 2^x + 5 - 3 \), analyze the behavior of the function as \( x \) approaches \( \pm \infty \).
A horizontal asymptote is a y-value (y = c) that the function approaches as \( x \) tends to \( \infty \) or \( -\infty \).
For \( f(x) = 2^x + 5 - 3 \):
- As \( x \to -\infty \), \( 2^x \to 0 \), thus \( f(x) \to 5 - 3 = 2 \).
- As \( x \to \infty \), \( 2^x \) grows exponentially, and thus the function does not approach a finite value.
Therefore, the only horizontal asymptote is \( y = 2 \). However, since neither of the provided options matches \( y = 2 \), the options seem to be incorrect or the question might be focusing on different types of asymptotes.
The horizontal asymptote described from the given function \( f(x) \) using these choices does not include the correct asymptote \( y = 2 \).
**Note to Students**
If you encounter discrepancies in multiple-choice options, make sure to re-evaluate the function and its behavior at the limits to identify errors and clarify with your instructor if needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c08cd6d-2150-4cc5-a139-43de6855ea2f%2Ff0e62f6c-0be6-41c4-8f98-2758bfc4764a%2Ft0qkofl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question
Which statement describes all the horizontal asymptotes of the function?
\[ f(x) = 2^x + 5 - 3 \]
- [ ] y = 0
- [ ] x = -3
- [ ] x = 0
- [ ] y = -3
**Explanation**
To identify the horizontal asymptotes of the function \( f(x) = 2^x + 5 - 3 \), analyze the behavior of the function as \( x \) approaches \( \pm \infty \).
A horizontal asymptote is a y-value (y = c) that the function approaches as \( x \) tends to \( \infty \) or \( -\infty \).
For \( f(x) = 2^x + 5 - 3 \):
- As \( x \to -\infty \), \( 2^x \to 0 \), thus \( f(x) \to 5 - 3 = 2 \).
- As \( x \to \infty \), \( 2^x \) grows exponentially, and thus the function does not approach a finite value.
Therefore, the only horizontal asymptote is \( y = 2 \). However, since neither of the provided options matches \( y = 2 \), the options seem to be incorrect or the question might be focusing on different types of asymptotes.
The horizontal asymptote described from the given function \( f(x) \) using these choices does not include the correct asymptote \( y = 2 \).
**Note to Students**
If you encounter discrepancies in multiple-choice options, make sure to re-evaluate the function and its behavior at the limits to identify errors and clarify with your instructor if needed.
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