At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To find the range of the function [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex] over the domain [tex]\( \{ x \mid -8 \leq x \leq 8 \} \)[/tex], follow these steps:
1. Understand the function:
The given function is [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex].
This tells us that for each [tex]\( x \)[/tex], [tex]\( y \)[/tex] is determined by taking the cube root of [tex]\(-x\)[/tex] and then subtracting 3.
2. Determine the domain:
The domain is [tex]\( \{ x \mid -8 \leq x \leq 8 \} \)[/tex].
3. Evaluate the cube root:
For different values of [tex]\( x \)[/tex] within the given domain, compute [tex]\( \sqrt[3]{-x} \)[/tex]:
- When [tex]\( x = -8 \)[/tex]: [tex]\( \sqrt[3]{-(-8)} = \sqrt[3]{8} = 2 \)[/tex]
- When [tex]\( x = 8 \)[/tex]: [tex]\( \sqrt[3]{-8} = \sqrt[3]{-8} = -2 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( \sqrt[3]{-0} = \sqrt[3]{0} = 0 \)[/tex]
4. Transform the results:
Now, apply the transformation [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex]:
- When [tex]\( x = -8 \)[/tex]: [tex]\( y = 2 - 3 = -1 \)[/tex]
- When [tex]\( x = 8 \)[/tex]: [tex]\( y = -2 - 3 = -5 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 0 - 3 = -3 \)[/tex]
5. Identify the range:
The critical points from above help outline the shape but checking the endpoints gives the values on both edges:
- For [tex]\( -8 \leq x \leq 8 \)[/tex], as [tex]\( x \)[/tex] increases from -8 to 8, the value of [tex]\( \sqrt[3]{-x} \)[/tex] smoothly decreases from [tex]\( +2 \)[/tex] to [tex]\( -2 \)[/tex].
- Thus [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex] smoothly decreases from [tex]\( -1 \)[/tex] to [tex]\( -5 \)[/tex].
This indicates the range of [tex]\( y \)[/tex] from highest to lowest values achieved.
6. Formulate the range:
From the calculations, [tex]\( y \)[/tex] ranges from the largest value -1 down to the smallest -5 as the cube root mapping covers all intermediate points' values:
- [tex]\(\min(y) = -5\)[/tex]
- [tex]\(\max(y) = -1\)[/tex]
Hence, the range of [tex]\( y \)[/tex] is:
[tex]\[ \{y \mid -5 \leq y \leq -1\} \][/tex]
So, the correct option is:
[tex]\[ \boxed{\{y \mid -5 \leq y \leq -1\}} \][/tex]
1. Understand the function:
The given function is [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex].
This tells us that for each [tex]\( x \)[/tex], [tex]\( y \)[/tex] is determined by taking the cube root of [tex]\(-x\)[/tex] and then subtracting 3.
2. Determine the domain:
The domain is [tex]\( \{ x \mid -8 \leq x \leq 8 \} \)[/tex].
3. Evaluate the cube root:
For different values of [tex]\( x \)[/tex] within the given domain, compute [tex]\( \sqrt[3]{-x} \)[/tex]:
- When [tex]\( x = -8 \)[/tex]: [tex]\( \sqrt[3]{-(-8)} = \sqrt[3]{8} = 2 \)[/tex]
- When [tex]\( x = 8 \)[/tex]: [tex]\( \sqrt[3]{-8} = \sqrt[3]{-8} = -2 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( \sqrt[3]{-0} = \sqrt[3]{0} = 0 \)[/tex]
4. Transform the results:
Now, apply the transformation [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex]:
- When [tex]\( x = -8 \)[/tex]: [tex]\( y = 2 - 3 = -1 \)[/tex]
- When [tex]\( x = 8 \)[/tex]: [tex]\( y = -2 - 3 = -5 \)[/tex]
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 0 - 3 = -3 \)[/tex]
5. Identify the range:
The critical points from above help outline the shape but checking the endpoints gives the values on both edges:
- For [tex]\( -8 \leq x \leq 8 \)[/tex], as [tex]\( x \)[/tex] increases from -8 to 8, the value of [tex]\( \sqrt[3]{-x} \)[/tex] smoothly decreases from [tex]\( +2 \)[/tex] to [tex]\( -2 \)[/tex].
- Thus [tex]\( y = \sqrt[3]{-x} - 3 \)[/tex] smoothly decreases from [tex]\( -1 \)[/tex] to [tex]\( -5 \)[/tex].
This indicates the range of [tex]\( y \)[/tex] from highest to lowest values achieved.
6. Formulate the range:
From the calculations, [tex]\( y \)[/tex] ranges from the largest value -1 down to the smallest -5 as the cube root mapping covers all intermediate points' values:
- [tex]\(\min(y) = -5\)[/tex]
- [tex]\(\max(y) = -1\)[/tex]
Hence, the range of [tex]\( y \)[/tex] is:
[tex]\[ \{y \mid -5 \leq y \leq -1\} \][/tex]
So, the correct option is:
[tex]\[ \boxed{\{y \mid -5 \leq y \leq -1\}} \][/tex]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.