Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Join our platform to connect with experts ready to provide accurate answers to your questions in various fields. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
To determine whether the equation [tex]\( x^2 + y^9 = 4 \)[/tex] defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], we first need to express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
### Step 1: Solve the equation for [tex]\( y \)[/tex]
We start with the given equation:
[tex]\[ x^2 + y^9 = 4 \][/tex]
To isolate [tex]\( y^9 \)[/tex], we subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[ y^9 = 4 - x^2 \][/tex]
Next, we solve for [tex]\( y \)[/tex] by taking the ninth root of both sides. This yields:
[tex]\[ y = \sqrt[9]{4 - x^2} \][/tex]
### Step 2: Consider the nature of the solutions
The equation [tex]\( y^9 = 4 - x^2 \)[/tex] implies that there are potentially multiple ninth roots for a given input [tex]\( x \)[/tex].
### Step 3: List the possible solutions for [tex]\( y \)[/tex]
The ninth root of a number can have multiple complex solutions. Specifically, for [tex]\( y = \sqrt[9]{4 - x^2} \)[/tex], the roots are:
1. [tex]\( y = (4 - x^2)^{1/9} \)[/tex]
2. [tex]\( y = -(4 - x^2)^{1/9} \cdot \cos(\pi/9) - i(4 - x^2)^{1/9} \cdot \sin(\pi/9) \)[/tex]
3. [tex]\( y = -(4 - x^2)^{1/9} \cdot \cos(\pi/9) + i(4 - x^2)^{1/9} \cdot \sin(\pi/9) \)[/tex]
4. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(2\pi/9) - i(4 - x^2)^{1/9} \cdot \sin(2\pi/9) \)[/tex]
5. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(2\pi/9) + i(4 - x^2)^{1/9} \cdot \sin(2\pi/9) \)[/tex]
6. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(4\pi/9) - i(4 - x^2)^{1/9} \cdot \sin(4\pi/9) \)[/tex]
7. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(4\pi/9) + i(4 - x^2)^{1/9} \cdot \sin(4\pi/9) \)[/tex]
8. [tex]\( y = -(4 - x^2)^{1/9} / 2 - i\sqrt{3}(4 - x^2)^{1/9} / 2 \)[/tex]
9. [tex]\( y = -(4 - x^2)^{1/9} / 2 + i\sqrt{3}(4 - x^2)^{1/9} / 2 \)[/tex]
### Step 4: Determine if [tex]\( y \)[/tex] is uniquely determined for a given [tex]\( x \)[/tex]
For [tex]\( y \)[/tex] to be a function of [tex]\( x \)[/tex], each input [tex]\( x \)[/tex] must correspond to exactly one output [tex]\( y \)[/tex]. From the list above, it is clear there are multiple possible values for [tex]\( y \)[/tex] for a given [tex]\( x \)[/tex].
### Conclusion
Since there are multiple solutions for [tex]\( y \)[/tex] for each value of [tex]\( x \)[/tex], the equation [tex]\( x^2 + y^9 = 4 \)[/tex] does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Correct Answer: The equation does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
### Step 1: Solve the equation for [tex]\( y \)[/tex]
We start with the given equation:
[tex]\[ x^2 + y^9 = 4 \][/tex]
To isolate [tex]\( y^9 \)[/tex], we subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[ y^9 = 4 - x^2 \][/tex]
Next, we solve for [tex]\( y \)[/tex] by taking the ninth root of both sides. This yields:
[tex]\[ y = \sqrt[9]{4 - x^2} \][/tex]
### Step 2: Consider the nature of the solutions
The equation [tex]\( y^9 = 4 - x^2 \)[/tex] implies that there are potentially multiple ninth roots for a given input [tex]\( x \)[/tex].
### Step 3: List the possible solutions for [tex]\( y \)[/tex]
The ninth root of a number can have multiple complex solutions. Specifically, for [tex]\( y = \sqrt[9]{4 - x^2} \)[/tex], the roots are:
1. [tex]\( y = (4 - x^2)^{1/9} \)[/tex]
2. [tex]\( y = -(4 - x^2)^{1/9} \cdot \cos(\pi/9) - i(4 - x^2)^{1/9} \cdot \sin(\pi/9) \)[/tex]
3. [tex]\( y = -(4 - x^2)^{1/9} \cdot \cos(\pi/9) + i(4 - x^2)^{1/9} \cdot \sin(\pi/9) \)[/tex]
4. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(2\pi/9) - i(4 - x^2)^{1/9} \cdot \sin(2\pi/9) \)[/tex]
5. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(2\pi/9) + i(4 - x^2)^{1/9} \cdot \sin(2\pi/9) \)[/tex]
6. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(4\pi/9) - i(4 - x^2)^{1/9} \cdot \sin(4\pi/9) \)[/tex]
7. [tex]\( y = (4 - x^2)^{1/9} \cdot \cos(4\pi/9) + i(4 - x^2)^{1/9} \cdot \sin(4\pi/9) \)[/tex]
8. [tex]\( y = -(4 - x^2)^{1/9} / 2 - i\sqrt{3}(4 - x^2)^{1/9} / 2 \)[/tex]
9. [tex]\( y = -(4 - x^2)^{1/9} / 2 + i\sqrt{3}(4 - x^2)^{1/9} / 2 \)[/tex]
### Step 4: Determine if [tex]\( y \)[/tex] is uniquely determined for a given [tex]\( x \)[/tex]
For [tex]\( y \)[/tex] to be a function of [tex]\( x \)[/tex], each input [tex]\( x \)[/tex] must correspond to exactly one output [tex]\( y \)[/tex]. From the list above, it is clear there are multiple possible values for [tex]\( y \)[/tex] for a given [tex]\( x \)[/tex].
### Conclusion
Since there are multiple solutions for [tex]\( y \)[/tex] for each value of [tex]\( x \)[/tex], the equation [tex]\( x^2 + y^9 = 4 \)[/tex] does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Correct Answer: The equation does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
