From science to arts, IDNLearn.com has the answers to all your questions. Our experts provide timely, comprehensive responses to ensure you have the information you need.
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
