Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine which function has the same range as [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex], we need to first evaluate the range of [tex]\( f(x) \)[/tex].
1. Evaluate the range of [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex]:
- The square root function [tex]\( \sqrt{x-3} \)[/tex] returns non-negative values because the square root of any real number is non-negative.
- Therefore, the minimum value of [tex]\( \sqrt{x-3} \)[/tex] is 0 when [tex]\( x = 3 \)[/tex], and it increases as [tex]\( x \)[/tex] increases.
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = -2 \sqrt{3-3} + 8 = -2 \cdot 0 + 8 = 8 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x-3} \)[/tex] increases, thus [tex]\( -2 \sqrt{x-3} \)[/tex] becomes more negative, decreasing the value of [tex]\( f(x) \)[/tex].
- Therefore, [tex]\( f(x) \)[/tex] starts at 8 and decreases without bound as [tex]\( x \)[/tex] increases.
- Thus, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 8] \)[/tex].
2. Evaluate the ranges of the given [tex]\( g(x) \)[/tex] functions:
- For [tex]\( g(x) = \sqrt{x-3} - 8 \)[/tex]:
- [tex]\( \sqrt{x-3} \geq 0 \)[/tex], so [tex]\( \sqrt{x-3} - 8 \geq -8 \)[/tex].
- Therefore, the range of this function is [tex]\([ -8, \infty )\)[/tex].
- For [tex]\( g(x) = \sqrt{x-3} + 8 \)[/tex]:
- [tex]\( \sqrt{x-3} \geq 0 \)[/tex], so [tex]\( \sqrt{x-3} + 8 \geq 8 \)[/tex].
- Therefore, the range of this function is [tex]\([ 8, \infty )\)[/tex].
- For [tex]\( g(x) = -\sqrt{x+3} + 8 \)[/tex]:
- The square root function [tex]\( \sqrt{x+3} \geq 0 \)[/tex], so the minimum value of [tex]\( \sqrt{x+3} \)[/tex] is when [tex]\( x = -3 \)[/tex].
- When [tex]\( x = -3 \)[/tex], [tex]\( g(x) = -\sqrt{-3+3} + 8 = -\sqrt{0} + 8 = 8 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x+3} \)[/tex] increases, thus [tex]\( -\sqrt{x+3} \)[/tex] decreases, making [tex]\( g(x) \)[/tex] decrease.
- Hence, the range of this function is [tex]\( (-\infty, 8] \)[/tex].
- For [tex]\( g(x) = -\sqrt{x-3} - 8 \)[/tex]:
- The square root function [tex]\( \sqrt{x-3} \geq 0 \)[/tex], so [tex]\( -\sqrt{x-3} \leq 0 \)[/tex].
- Therefore, the maximum value of [tex]\( g(x) = -\sqrt{x-3} - 8 \)[/tex] occurs when [tex]\( \sqrt{x-3}=0 \)[/tex], which gives [tex]\( g(x) = -8 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x-3} \)[/tex] increases, making [tex]\( -\sqrt{x-3} \)[/tex] more negative.
- Hence, the range of this function is [tex]\( (-\infty, -8 ]\)[/tex].
Upon analyzing the ranges, we can see that the function [tex]\( g(x) = -\sqrt{x+3} + 8 \)[/tex] has the same range as [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex], which is [tex]\( (-\infty, 8] \)[/tex].
Thus, the function that has the same range as [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex] is [tex]\( \boxed{-\sqrt{x+3} + 8} \)[/tex].
1. Evaluate the range of [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex]:
- The square root function [tex]\( \sqrt{x-3} \)[/tex] returns non-negative values because the square root of any real number is non-negative.
- Therefore, the minimum value of [tex]\( \sqrt{x-3} \)[/tex] is 0 when [tex]\( x = 3 \)[/tex], and it increases as [tex]\( x \)[/tex] increases.
- When [tex]\( x = 3 \)[/tex], [tex]\( f(x) = -2 \sqrt{3-3} + 8 = -2 \cdot 0 + 8 = 8 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x-3} \)[/tex] increases, thus [tex]\( -2 \sqrt{x-3} \)[/tex] becomes more negative, decreasing the value of [tex]\( f(x) \)[/tex].
- Therefore, [tex]\( f(x) \)[/tex] starts at 8 and decreases without bound as [tex]\( x \)[/tex] increases.
- Thus, the range of [tex]\( f(x) \)[/tex] is [tex]\( (-\infty, 8] \)[/tex].
2. Evaluate the ranges of the given [tex]\( g(x) \)[/tex] functions:
- For [tex]\( g(x) = \sqrt{x-3} - 8 \)[/tex]:
- [tex]\( \sqrt{x-3} \geq 0 \)[/tex], so [tex]\( \sqrt{x-3} - 8 \geq -8 \)[/tex].
- Therefore, the range of this function is [tex]\([ -8, \infty )\)[/tex].
- For [tex]\( g(x) = \sqrt{x-3} + 8 \)[/tex]:
- [tex]\( \sqrt{x-3} \geq 0 \)[/tex], so [tex]\( \sqrt{x-3} + 8 \geq 8 \)[/tex].
- Therefore, the range of this function is [tex]\([ 8, \infty )\)[/tex].
- For [tex]\( g(x) = -\sqrt{x+3} + 8 \)[/tex]:
- The square root function [tex]\( \sqrt{x+3} \geq 0 \)[/tex], so the minimum value of [tex]\( \sqrt{x+3} \)[/tex] is when [tex]\( x = -3 \)[/tex].
- When [tex]\( x = -3 \)[/tex], [tex]\( g(x) = -\sqrt{-3+3} + 8 = -\sqrt{0} + 8 = 8 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x+3} \)[/tex] increases, thus [tex]\( -\sqrt{x+3} \)[/tex] decreases, making [tex]\( g(x) \)[/tex] decrease.
- Hence, the range of this function is [tex]\( (-\infty, 8] \)[/tex].
- For [tex]\( g(x) = -\sqrt{x-3} - 8 \)[/tex]:
- The square root function [tex]\( \sqrt{x-3} \geq 0 \)[/tex], so [tex]\( -\sqrt{x-3} \leq 0 \)[/tex].
- Therefore, the maximum value of [tex]\( g(x) = -\sqrt{x-3} - 8 \)[/tex] occurs when [tex]\( \sqrt{x-3}=0 \)[/tex], which gives [tex]\( g(x) = -8 \)[/tex].
- As [tex]\( x \)[/tex] increases, [tex]\( \sqrt{x-3} \)[/tex] increases, making [tex]\( -\sqrt{x-3} \)[/tex] more negative.
- Hence, the range of this function is [tex]\( (-\infty, -8 ]\)[/tex].
Upon analyzing the ranges, we can see that the function [tex]\( g(x) = -\sqrt{x+3} + 8 \)[/tex] has the same range as [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex], which is [tex]\( (-\infty, 8] \)[/tex].
Thus, the function that has the same range as [tex]\( f(x) = -2 \sqrt{x-3} + 8 \)[/tex] is [tex]\( \boxed{-\sqrt{x+3} + 8} \)[/tex].
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.
