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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.
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