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].
