Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine which statement best describes the function [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex] in relation to the value [tex]\(-6\)[/tex], let's carefully analyze the domain and range of the function and the given value.
1. Domain Analysis:
The domain of [tex]\( f(x) \)[/tex] is determined by the expression inside the square root [tex]\(\sqrt{x-7}\)[/tex]. For the square root to be defined, the expression inside must be non-negative:
[tex]\[ x - 7 \geq 0 \implies x \geq 7 \][/tex]
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( [7, \infty) \)[/tex].
2. Range Analysis:
To find the range of [tex]\( f(x) \)[/tex], we evaluate the bounds of the function:
- At the lower boundary of the domain ([tex]\( x = 7 \)[/tex]), the function value is:
[tex]\[ f(7) = -2 \sqrt{7-7} + 1 = -2 \cdot 0 + 1 = 1 \][/tex]
- As [tex]\( x \to \infty \)[/tex], [tex]\( \sqrt{x-7} \to \infty \)[/tex] and hence [tex]\( -2 \sqrt{x-7} \to -\infty \)[/tex], making [tex]\( f(x) \)[/tex] approach [tex]\( -\infty \)[/tex].
Therefore, the range is [tex]\( (-\infty, 1] \)[/tex].
3. Checking [tex]\(-6\)[/tex]:
- Domain Check:
[tex]\(-6 < 7\)[/tex], which means [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
- Range Check:
We need to check if [tex]\(-6\)[/tex] can be an output of [tex]\( f(x) \)[/tex]. Thus, we solve for [tex]\( x \)[/tex] when [tex]\( f(x) = -6 \)[/tex]:
[tex]\[ -6 = -2 \sqrt{x-7} + 1 \][/tex]
Rearranging the equation:
[tex]\[ -6 - 1 = -2 \sqrt{x-7} \implies -7 = -2 \sqrt{x-7} \implies 7 = 2 \sqrt{x-7} \implies \sqrt{x-7} = \frac{7}{2} \implies x-7 = \left( \frac{7}{2} \right)^2 = \frac{49}{4} \implies x = 7 + \frac{49}{4} = \frac{77}{4} \][/tex]
Since [tex]\( x = \frac{77}{4} = 19.25 \geq 7 \)[/tex], [tex]\(-6\)[/tex] is indeed in the range of [tex]\( f(x) \)[/tex].
Combining the results:
- [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
- [tex]\(-6\)[/tex] is in the range of [tex]\( f(x) \)[/tex].
Therefore, the best statement describing the function [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex] is:
-6 is not in the domain of [tex]\( f(x) \)[/tex] but is in the range of [tex]\( f(x) \)[/tex].
1. Domain Analysis:
The domain of [tex]\( f(x) \)[/tex] is determined by the expression inside the square root [tex]\(\sqrt{x-7}\)[/tex]. For the square root to be defined, the expression inside must be non-negative:
[tex]\[ x - 7 \geq 0 \implies x \geq 7 \][/tex]
Therefore, the domain of [tex]\( f(x) \)[/tex] is [tex]\( [7, \infty) \)[/tex].
2. Range Analysis:
To find the range of [tex]\( f(x) \)[/tex], we evaluate the bounds of the function:
- At the lower boundary of the domain ([tex]\( x = 7 \)[/tex]), the function value is:
[tex]\[ f(7) = -2 \sqrt{7-7} + 1 = -2 \cdot 0 + 1 = 1 \][/tex]
- As [tex]\( x \to \infty \)[/tex], [tex]\( \sqrt{x-7} \to \infty \)[/tex] and hence [tex]\( -2 \sqrt{x-7} \to -\infty \)[/tex], making [tex]\( f(x) \)[/tex] approach [tex]\( -\infty \)[/tex].
Therefore, the range is [tex]\( (-\infty, 1] \)[/tex].
3. Checking [tex]\(-6\)[/tex]:
- Domain Check:
[tex]\(-6 < 7\)[/tex], which means [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
- Range Check:
We need to check if [tex]\(-6\)[/tex] can be an output of [tex]\( f(x) \)[/tex]. Thus, we solve for [tex]\( x \)[/tex] when [tex]\( f(x) = -6 \)[/tex]:
[tex]\[ -6 = -2 \sqrt{x-7} + 1 \][/tex]
Rearranging the equation:
[tex]\[ -6 - 1 = -2 \sqrt{x-7} \implies -7 = -2 \sqrt{x-7} \implies 7 = 2 \sqrt{x-7} \implies \sqrt{x-7} = \frac{7}{2} \implies x-7 = \left( \frac{7}{2} \right)^2 = \frac{49}{4} \implies x = 7 + \frac{49}{4} = \frac{77}{4} \][/tex]
Since [tex]\( x = \frac{77}{4} = 19.25 \geq 7 \)[/tex], [tex]\(-6\)[/tex] is indeed in the range of [tex]\( f(x) \)[/tex].
Combining the results:
- [tex]\(-6\)[/tex] is not in the domain of [tex]\( f(x) \)[/tex].
- [tex]\(-6\)[/tex] is in the range of [tex]\( f(x) \)[/tex].
Therefore, the best statement describing the function [tex]\( f(x) = -2 \sqrt{x-7} + 1 \)[/tex] is:
-6 is not in the domain of [tex]\( f(x) \)[/tex] but is in the range of [tex]\( f(x) \)[/tex].
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
