Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Connect with a community of experts ready to help you find accurate solutions to your questions quickly and efficiently. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine the end behavior of the function [tex]\( g(x) = 4|x-2| - 3 \)[/tex], let's analyze the function as [tex]\( x \)[/tex] approaches negative and positive infinity.
1. As [tex]\( x \)[/tex] approaches negative infinity:
When [tex]\( x \)[/tex] is a very large negative number, the term [tex]\(|x-2|\)[/tex] will also be a large positive number since the absolute value of a large negative number shifted by 2 is still large. Therefore, [tex]\(|x-2| \approx |x|\)[/tex] and thus behaves like [tex]\( |x| \)[/tex]. Consequently,
[tex]\[ g(x) \approx 4|x| - 3. \][/tex]
Since [tex]\( |x| \)[/tex] equals [tex]\( -x \)[/tex] when [tex]\( x \)[/tex] is negative, we have:
[tex]\[ |x-2| \approx -x \][/tex]
Thus,
[tex]\[ g(x) \approx 4(-x) - 3 = -4x - 3. \][/tex]
As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( -4x - 3 \)[/tex] will decrease without bound, approaching negative infinity.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches negative infinity.
2. As [tex]\( x \)[/tex] approaches positive infinity:
When [tex]\( x \)[/tex] is a very large positive number, the term [tex]\(|x-2|\)[/tex] will be approximately equal to [tex]\( x \)[/tex] itself, because the shift by 2 becomes negligible for very large values of [tex]\( x \)[/tex]. Thus,
[tex]\[ |x-2| \approx x. \][/tex]
Then,
[tex]\[ g(x) \approx 4x - 3. \][/tex]
As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 4x - 3 \)[/tex] will increase without bound, approaching positive infinity.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
In summary, the correct selections for the function [tex]\( g(x) = 4|x-2| - 3 \)[/tex] are:
As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches negative infinity.
As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
1. As [tex]\( x \)[/tex] approaches negative infinity:
When [tex]\( x \)[/tex] is a very large negative number, the term [tex]\(|x-2|\)[/tex] will also be a large positive number since the absolute value of a large negative number shifted by 2 is still large. Therefore, [tex]\(|x-2| \approx |x|\)[/tex] and thus behaves like [tex]\( |x| \)[/tex]. Consequently,
[tex]\[ g(x) \approx 4|x| - 3. \][/tex]
Since [tex]\( |x| \)[/tex] equals [tex]\( -x \)[/tex] when [tex]\( x \)[/tex] is negative, we have:
[tex]\[ |x-2| \approx -x \][/tex]
Thus,
[tex]\[ g(x) \approx 4(-x) - 3 = -4x - 3. \][/tex]
As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( -4x - 3 \)[/tex] will decrease without bound, approaching negative infinity.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches negative infinity.
2. As [tex]\( x \)[/tex] approaches positive infinity:
When [tex]\( x \)[/tex] is a very large positive number, the term [tex]\(|x-2|\)[/tex] will be approximately equal to [tex]\( x \)[/tex] itself, because the shift by 2 becomes negligible for very large values of [tex]\( x \)[/tex]. Thus,
[tex]\[ |x-2| \approx x. \][/tex]
Then,
[tex]\[ g(x) \approx 4x - 3. \][/tex]
As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( 4x - 3 \)[/tex] will increase without bound, approaching positive infinity.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
In summary, the correct selections for the function [tex]\( g(x) = 4|x-2| - 3 \)[/tex] are:
As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( g(x) \)[/tex] approaches negative infinity.
As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( g(x) \)[/tex] approaches positive infinity.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.