Discover new perspectives and gain insights with IDNLearn.com's diverse answers. Our Q&A platform offers reliable and thorough answers to help you make informed decisions quickly and easily.
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].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
