To determine whether the given equation
[tex]\[ 8x^2 - 9y^2 = 1 \][/tex]
defines [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex], let’s follow these steps:
1. Rearrange the equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 8x^2 - 1 = 9y^2 \][/tex]
Dividing both sides by 9 gives:
[tex]\[ y^2 = \frac{8x^2 - 1}{9} \][/tex]
Taking the square root of both sides, we get:
[tex]\[ y = \pm \sqrt{\frac{8x^2 - 1}{9}} \][/tex]
2. Analyze the outputs for a given input [tex]\( x \)[/tex]:
The equation [tex]\[ y = \pm \sqrt{\frac{8x^2 - 1}{9}} \][/tex] indicates that for any given [tex]\( x \)[/tex] within the domain where the expression inside the square root is non-negative, there will be two possible values of [tex]\( y \)[/tex]: a positive value and a negative value.
3. Determine if [tex]\( y \)[/tex] is a function of [tex]\( x \)[/tex]:
By definition, a function can only have one output for each input. Here, for any [tex]\( x \)[/tex] within the appropriate domain, there are two outputs: [tex]\( \sqrt{\frac{8x^2 - 1}{9}} \)[/tex] and [tex]\( -\sqrt{\frac{8x^2 - 1}{9}} \)[/tex].
Since there are two different [tex]\( y \)[/tex] values for a single [tex]\( x \)[/tex], the equation does not satisfy the requirement of defining [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex].
Therefore, the correct answer is:
D. The equation does not define [tex]\( y \)[/tex] as a function of [tex]\( x \)[/tex] because for at least one input value of [tex]\( x \)[/tex], in the domain, the equation yields two different outputs.
