Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
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].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
