Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine which transformation of the parent square root function results in the given domain of [tex]\([2, \infty)\)[/tex] and range of [tex]\([3, \infty)\)[/tex], we need to examine each of the provided functions.
The parent square root function is [tex]\( f(x) = \sqrt{x} \)[/tex].
### A. [tex]\( j(x) = \sqrt{x + 2} + 3 \)[/tex]
1. Domain: For [tex]\( j(x) \)[/tex], [tex]\( x + 2 \geq 0 \Rightarrow x \geq -2 \)[/tex]. So, the domain is [tex]\([ -2, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x + 2} + 3\)[/tex] occurs when [tex]\(x = -2\)[/tex]:
[tex]\[ j(-2) = \sqrt{-2 + 2} + 3 = 0 + 3 = 3 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x + 2} \)[/tex] increases without bound. Thus, the range is [tex]\([3, \infty)\)[/tex].
The range matches the given range [tex]\([3, \infty)\)[/tex], but the domain [tex]\([ -2, \infty)\)[/tex] does not match the given domain [tex]\([2, \infty)\)[/tex]. Therefore, option A is not correct.
### B. [tex]\( k(x) = \sqrt{x + 3} - 2 \)[/tex]
1. Domain: For [tex]\( k(x) \)[/tex], [tex]\( x + 3 \geq 0 \Rightarrow x \geq -3 \)[/tex]. So, the domain is [tex]\([ -3, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x + 3} - 2\)[/tex] occurs when [tex]\(x = -3\)[/tex]:
[tex]\[ k(-3) = \sqrt{-3 + 3} - 2 = 0 - 2 = -2 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x + 3} \)[/tex] increases without bound. Thus, the range is [tex]\([-2, \infty)\)[/tex].
Neither the domain nor the range matches the given constraints. Therefore, option B is not correct.
### C. [tex]\( h(x) = \sqrt{x - 3} - 2 \)[/tex]
1. Domain: For [tex]\( h(x) \)[/tex], [tex]\( x - 3 \geq 0 \Rightarrow x \geq 3 \)[/tex]. So, the domain is [tex]\([3, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x - 3} - 2\)[/tex] occurs when [tex]\(x = 3\)[/tex]:
[tex]\[ h(3) = \sqrt{3 - 3} - 2 = 0 - 2 = -2 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x - 3} \)[/tex] increases without bound. Thus, the range is [tex]\([-2, \infty)\)[/tex].
The domain matches the given domain [tex]\([3, \infty)\)[/tex], but the range [tex]\([-2, \infty)\)[/tex] does not match the given range [tex]\([3, \infty)\)[/tex]. Therefore, option C is not correct.
### D. [tex]\( g(x) = \sqrt{x - 2} + 3 \)[/tex]
1. Domain: For [tex]\( g(x) \)[/tex], [tex]\( x - 2 \geq 0 \Rightarrow x \geq 2 \)[/tex]. So, the domain is [tex]\([2, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x - 2} + 3\)[/tex] occurs when [tex]\(x = 2\)[/tex]:
[tex]\[ g(2) = \sqrt{2 - 2} + 3 = 0 + 3 = 3 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x - 2} \)[/tex] increases without bound. Thus, the range is [tex]\([3, \infty)\)[/tex].
Both the domain [tex]\([2, \infty)\)[/tex] and the range [tex]\([3, \infty)\)[/tex] match the given conditions. Therefore, the correct transformation is:
D. [tex]\( g(x) = \sqrt{x - 2} + 3 \)[/tex]
Thus, the answer is D.
The parent square root function is [tex]\( f(x) = \sqrt{x} \)[/tex].
### A. [tex]\( j(x) = \sqrt{x + 2} + 3 \)[/tex]
1. Domain: For [tex]\( j(x) \)[/tex], [tex]\( x + 2 \geq 0 \Rightarrow x \geq -2 \)[/tex]. So, the domain is [tex]\([ -2, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x + 2} + 3\)[/tex] occurs when [tex]\(x = -2\)[/tex]:
[tex]\[ j(-2) = \sqrt{-2 + 2} + 3 = 0 + 3 = 3 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x + 2} \)[/tex] increases without bound. Thus, the range is [tex]\([3, \infty)\)[/tex].
The range matches the given range [tex]\([3, \infty)\)[/tex], but the domain [tex]\([ -2, \infty)\)[/tex] does not match the given domain [tex]\([2, \infty)\)[/tex]. Therefore, option A is not correct.
### B. [tex]\( k(x) = \sqrt{x + 3} - 2 \)[/tex]
1. Domain: For [tex]\( k(x) \)[/tex], [tex]\( x + 3 \geq 0 \Rightarrow x \geq -3 \)[/tex]. So, the domain is [tex]\([ -3, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x + 3} - 2\)[/tex] occurs when [tex]\(x = -3\)[/tex]:
[tex]\[ k(-3) = \sqrt{-3 + 3} - 2 = 0 - 2 = -2 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x + 3} \)[/tex] increases without bound. Thus, the range is [tex]\([-2, \infty)\)[/tex].
Neither the domain nor the range matches the given constraints. Therefore, option B is not correct.
### C. [tex]\( h(x) = \sqrt{x - 3} - 2 \)[/tex]
1. Domain: For [tex]\( h(x) \)[/tex], [tex]\( x - 3 \geq 0 \Rightarrow x \geq 3 \)[/tex]. So, the domain is [tex]\([3, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x - 3} - 2\)[/tex] occurs when [tex]\(x = 3\)[/tex]:
[tex]\[ h(3) = \sqrt{3 - 3} - 2 = 0 - 2 = -2 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x - 3} \)[/tex] increases without bound. Thus, the range is [tex]\([-2, \infty)\)[/tex].
The domain matches the given domain [tex]\([3, \infty)\)[/tex], but the range [tex]\([-2, \infty)\)[/tex] does not match the given range [tex]\([3, \infty)\)[/tex]. Therefore, option C is not correct.
### D. [tex]\( g(x) = \sqrt{x - 2} + 3 \)[/tex]
1. Domain: For [tex]\( g(x) \)[/tex], [tex]\( x - 2 \geq 0 \Rightarrow x \geq 2 \)[/tex]. So, the domain is [tex]\([2, \infty )\)[/tex].
2. Range: The minimum value of [tex]\(\sqrt{x - 2} + 3\)[/tex] occurs when [tex]\(x = 2\)[/tex]:
[tex]\[ g(2) = \sqrt{2 - 2} + 3 = 0 + 3 = 3 \][/tex]
As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x - 2} \)[/tex] increases without bound. Thus, the range is [tex]\([3, \infty)\)[/tex].
Both the domain [tex]\([2, \infty)\)[/tex] and the range [tex]\([3, \infty)\)[/tex] match the given conditions. Therefore, the correct transformation is:
D. [tex]\( g(x) = \sqrt{x - 2} + 3 \)[/tex]
Thus, the answer is D.
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
