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Given f(x)= › =(-³)* and g(x) = -3x - 1, which of the following accurately describes the end behavior the two graphs have in common? O The end behavior on both sides is the same. O Only the right end behavior is the same. O Only the left end behavior is the same. Neither end behavior is the same.

Given f(x)= › =(-³)* and g(x) = -3x - 1, which of the following accurately describes the end behavior the two graphs have in common? O The end behavior on both sides is the same. O Only the right end behavior is the same. O Only the left end behavior is the same. Neither end behavior is the same.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Given \( f(x) = \left(\frac{1}{3}\right)^x \) and \( g(x) = -3x - 1 \), which of the following accurately describes the end behavior the two graphs have in common? **Options:** - ○ The end behavior on both sides is the same. - ○ Only the right end behavior is the same. - ○ Only the left end behavior is the same. - ○ Neither end behavior is the same. **Explanation:** The problem asks which of the given options correctly identifies the shared end behavior, if any, between the two functions \( f(x) = \left(\frac{1}{3}\right)^x \) and \( g(x) = -3x - 1 \). Analyzing the given functions: 1. **Exponential Function:** \( f(x) = \left(\frac{1}{3}\right)^x \) - As \( x \to \infty \), \( f(x) \to 0 \). - As \( x \to -\infty \), \( f(x) \to \infty \). 2. **Linear Function:** \( g(x) = -3x - 1 \) - As \( x \to \infty \), \( g(x) \to -\infty \). - As \( x \to -\infty \), \( g(x) \to \infty \). In summary, both \( f(x) \) and \( g(x) \) share the same left end behavior, tending towards infinity as \( x \to -\infty \), but have different behaviors as \( x \to \infty \). Therefore, the correct choice is: - ○ Only the left end behavior is the same.
Transcribed Image Text:**Problem Statement:** Given \( f(x) = \left(\frac{1}{3}\right)^x \) and \( g(x) = -3x - 1 \), which of the following accurately describes the end behavior the two graphs have in common? **Options:** - ○ The end behavior on both sides is the same. - ○ Only the right end behavior is the same. - ○ Only the left end behavior is the same. - ○ Neither end behavior is the same. **Explanation:** The problem asks which of the given options correctly identifies the shared end behavior, if any, between the two functions \( f(x) = \left(\frac{1}{3}\right)^x \) and \( g(x) = -3x - 1 \). Analyzing the given functions: 1. **Exponential Function:** \( f(x) = \left(\frac{1}{3}\right)^x \) - As \( x \to \infty \), \( f(x) \to 0 \). - As \( x \to -\infty \), \( f(x) \to \infty \). 2. **Linear Function:** \( g(x) = -3x - 1 \) - As \( x \to \infty \), \( g(x) \to -\infty \). - As \( x \to -\infty \), \( g(x) \to \infty \). In summary, both \( f(x) \) and \( g(x) \) share the same left end behavior, tending towards infinity as \( x \to -\infty \), but have different behaviors as \( x \to \infty \). Therefore, the correct choice is: - ○ Only the left end behavior is the same.
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