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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]
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