Discover a world of knowledge at Westonci.ca, where experts and enthusiasts come together to answer your questions. Get detailed and accurate answers to your questions from a community of experts on our comprehensive Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
Let’s analyze the end behavior of the function [tex]\( g(x) = \frac{x^2 + 6x}{30} \)[/tex].
Step-by-Step Analysis:
1. Understanding the Function: We have [tex]\( g(x) = \frac{x^2 + 6x}{30} \)[/tex].
2. Leading Term Dominance: For large values of [tex]\( x \)[/tex] (either very large positive or very large negative), the [tex]\( x^2 \)[/tex] term in the numerator will dominate over the [tex]\( 6x \)[/tex] term since [tex]\( x^2 \)[/tex] grows much faster than [tex]\( x \)[/tex].
3. End Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large, [tex]\( x^2 \)[/tex] will dominate, and [tex]\( g(x) \)[/tex] will behave approximately like [tex]\( \frac{x^2}{30} \)[/tex].
- Thus, as [tex]\( x \to \infty \)[/tex], [tex]\( \frac{x^2}{30} \)[/tex] increases without bound.
- Therefore, [tex]\( g(x) \to \infty \)[/tex] as [tex]\( x \to \infty \)[/tex].
4. End Behavior as [tex]\( x \to -\infty \)[/tex]:
- Similarly, for very large negative values of [tex]\( x \)[/tex], [tex]\( x^2 \)[/tex] still dominates because [tex]\( (-x)^2 = x^2 \)[/tex].
- So, [tex]\( g(x) \)[/tex] will again behave like [tex]\( \frac{x^2}{30} \)[/tex].
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( \frac{x^2}{30} \)[/tex] increases without bound.
- Thus, [tex]\( g(x) \to \infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
From this analysis, we can conclude that as [tex]\( x \)[/tex] approaches either [tex]\( -\infty \)[/tex] or [tex]\( \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
Selecting Correct Statement:
Given these conclusions, the correct statement is:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex]; and as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
However, this exact match is not provided in the options. The closest option, aligning to our conclusion, is the interpretation of the positive bound behavior:
The closest correct option seems to be:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] or [tex]\( \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
Which aligns with one intermediate result as:
So, reflecting on the best available correct behavior:
Correct Answer: As [tex]\( x \)[/tex] approaches [tex]\( -\infty, g(x) \)[/tex] approaches [tex]\( \infty \)[/tex]; and as [tex]\( x \)[/tex] approaches [tex]\( \infty, g(x) approaches \infty \)[/tex].
Step-by-Step Analysis:
1. Understanding the Function: We have [tex]\( g(x) = \frac{x^2 + 6x}{30} \)[/tex].
2. Leading Term Dominance: For large values of [tex]\( x \)[/tex] (either very large positive or very large negative), the [tex]\( x^2 \)[/tex] term in the numerator will dominate over the [tex]\( 6x \)[/tex] term since [tex]\( x^2 \)[/tex] grows much faster than [tex]\( x \)[/tex].
3. End Behavior as [tex]\( x \to \infty \)[/tex]:
- As [tex]\( x \)[/tex] becomes very large, [tex]\( x^2 \)[/tex] will dominate, and [tex]\( g(x) \)[/tex] will behave approximately like [tex]\( \frac{x^2}{30} \)[/tex].
- Thus, as [tex]\( x \to \infty \)[/tex], [tex]\( \frac{x^2}{30} \)[/tex] increases without bound.
- Therefore, [tex]\( g(x) \to \infty \)[/tex] as [tex]\( x \to \infty \)[/tex].
4. End Behavior as [tex]\( x \to -\infty \)[/tex]:
- Similarly, for very large negative values of [tex]\( x \)[/tex], [tex]\( x^2 \)[/tex] still dominates because [tex]\( (-x)^2 = x^2 \)[/tex].
- So, [tex]\( g(x) \)[/tex] will again behave like [tex]\( \frac{x^2}{30} \)[/tex].
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( \frac{x^2}{30} \)[/tex] increases without bound.
- Thus, [tex]\( g(x) \to \infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
From this analysis, we can conclude that as [tex]\( x \)[/tex] approaches either [tex]\( -\infty \)[/tex] or [tex]\( \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
Selecting Correct Statement:
Given these conclusions, the correct statement is:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex]; and as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
However, this exact match is not provided in the options. The closest option, aligning to our conclusion, is the interpretation of the positive bound behavior:
The closest correct option seems to be:
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] or [tex]\( \infty \)[/tex], [tex]\( g(x) \)[/tex] approaches [tex]\( \infty \)[/tex].
Which aligns with one intermediate result as:
So, reflecting on the best available correct behavior:
Correct Answer: As [tex]\( x \)[/tex] approaches [tex]\( -\infty, g(x) \)[/tex] approaches [tex]\( \infty \)[/tex]; and as [tex]\( x \)[/tex] approaches [tex]\( \infty, g(x) approaches \infty \)[/tex].
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
