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To solve the problem given the equation [tex]\( y = -\sqrt[3]{x} + 4 \)[/tex], we need to evaluate [tex]\( y \)[/tex] for different values of [tex]\( x \)[/tex]. We’ll consider a set of [tex]\( x \)[/tex]-values and compute the corresponding [tex]\( y \)[/tex]-values.
Let's choose the following values of [tex]\( x \)[/tex]:
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = 8 \)[/tex]
- [tex]\( x = 27 \)[/tex]
- [tex]\( x = 64 \)[/tex]
- [tex]\( x = 125 \)[/tex]
We will now calculate the [tex]\( y \)[/tex]-values for each of these [tex]\( x \)[/tex]-values step-by-step.
### Step-by-Step Calculation
1. For [tex]\( x = 1 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{1} \)[/tex] which is [tex]\( 1 \)[/tex]
- Compute [tex]\( y = -1 + 4 \)[/tex]
- Result: [tex]\( y = 3 \)[/tex]
2. For [tex]\( x = 8 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{8} \)[/tex] which is [tex]\( 2 \)[/tex]
- Compute [tex]\( y = -2 + 4 \)[/tex]
- Result: [tex]\( y = 2 \)[/tex]
3. For [tex]\( x = 27 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{27} \)[/tex] which is [tex]\( 3 \)[/tex]
- Compute [tex]\( y = -3 + 4 \)[/tex]
- Result: [tex]\( y = 1 \)[/tex]
4. For [tex]\( x = 64 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{64} \)[/tex] which is [tex]\( 4 \)[/tex]
- Compute [tex]\( y = -4 + 4 \)[/tex]
- Result: [tex]\( y \approx 4.440892098500626 \times 10^{-16} \)[/tex]
5. For [tex]\( x = 125 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{125} \)[/tex] which is [tex]\( 5 \)[/tex]
- Compute [tex]\( y = -5 + 4 \)[/tex]
- Result: [tex]\( y \approx -0.9999999999999991 \)[/tex]
### Results Summary
For the chosen [tex]\( x \)[/tex]-values, the corresponding [tex]\( y \)[/tex]-values are:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex]
- When [tex]\( x = 8 \)[/tex], [tex]\( y = 2 \)[/tex]
- When [tex]\( x = 27 \)[/tex], [tex]\( y = 1 \)[/tex]
- When [tex]\( x = 64 \)[/tex], [tex]\( y \approx 4.440892098500626 \times 10^{-16} \)[/tex]
- When [tex]\( x = 125 \)[/tex], [tex]\( y \approx -0.9999999999999991 \)[/tex]
These values help us understand how the function [tex]\( y = -\sqrt[3]{x} + 4 \)[/tex] behaves for increasing values of [tex]\( x \)[/tex].
Let's choose the following values of [tex]\( x \)[/tex]:
- [tex]\( x = 1 \)[/tex]
- [tex]\( x = 8 \)[/tex]
- [tex]\( x = 27 \)[/tex]
- [tex]\( x = 64 \)[/tex]
- [tex]\( x = 125 \)[/tex]
We will now calculate the [tex]\( y \)[/tex]-values for each of these [tex]\( x \)[/tex]-values step-by-step.
### Step-by-Step Calculation
1. For [tex]\( x = 1 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{1} \)[/tex] which is [tex]\( 1 \)[/tex]
- Compute [tex]\( y = -1 + 4 \)[/tex]
- Result: [tex]\( y = 3 \)[/tex]
2. For [tex]\( x = 8 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{8} \)[/tex] which is [tex]\( 2 \)[/tex]
- Compute [tex]\( y = -2 + 4 \)[/tex]
- Result: [tex]\( y = 2 \)[/tex]
3. For [tex]\( x = 27 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{27} \)[/tex] which is [tex]\( 3 \)[/tex]
- Compute [tex]\( y = -3 + 4 \)[/tex]
- Result: [tex]\( y = 1 \)[/tex]
4. For [tex]\( x = 64 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{64} \)[/tex] which is [tex]\( 4 \)[/tex]
- Compute [tex]\( y = -4 + 4 \)[/tex]
- Result: [tex]\( y \approx 4.440892098500626 \times 10^{-16} \)[/tex]
5. For [tex]\( x = 125 \)[/tex]:
- Calculate [tex]\( \sqrt[3]{125} \)[/tex] which is [tex]\( 5 \)[/tex]
- Compute [tex]\( y = -5 + 4 \)[/tex]
- Result: [tex]\( y \approx -0.9999999999999991 \)[/tex]
### Results Summary
For the chosen [tex]\( x \)[/tex]-values, the corresponding [tex]\( y \)[/tex]-values are:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 3 \)[/tex]
- When [tex]\( x = 8 \)[/tex], [tex]\( y = 2 \)[/tex]
- When [tex]\( x = 27 \)[/tex], [tex]\( y = 1 \)[/tex]
- When [tex]\( x = 64 \)[/tex], [tex]\( y \approx 4.440892098500626 \times 10^{-16} \)[/tex]
- When [tex]\( x = 125 \)[/tex], [tex]\( y \approx -0.9999999999999991 \)[/tex]
These values help us understand how the function [tex]\( y = -\sqrt[3]{x} + 4 \)[/tex] behaves for increasing values of [tex]\( x \)[/tex].
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