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].
