To find the inverse of the equation [tex]\( y = 2x^2 \)[/tex], we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]. Follow these steps to derive the inverse:
1. Start with the original equation:
[tex]\[
y = 2x^2
\][/tex]
2. Solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{y}{2}
\][/tex]
3. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{y}{2}}
\][/tex]
Simplifying further:
[tex]\[
x = \pm \frac{\sqrt{2y}}{2}
\][/tex]
This means we have two solutions, [tex]\( x = \frac{\sqrt{2y}}{2} \)[/tex] and [tex]\( x = -\frac{\sqrt{2y}}{2} \)[/tex].
4. Substitute back into the context to understand the inverse relationship:
[tex]\[
x = \pm \frac{\sqrt{2y}}{2}
\][/tex]
Therefore, the equation that correctly represents the inverse of [tex]\( y = 2x^2 \)[/tex] when simplified would be:
[tex]\[
\boxed{x = \pm \frac{\sqrt{2y}}{2}}
\][/tex]
Among the given options, the first three options can be rearranged and simplified, but none of them directly provide the inverse relationship. The fourth option [tex]\( x = 2y^2 \)[/tex] is incorrect since it does not align with our derived inverse. Thus, based on our derived step-by-step solution, we identify the simplified form for the inverse relationship. No given option correctly simplifies to the inverse; hence, the derived result ([tex]\( x = \pm \frac{\sqrt{2y}}{2} \)[/tex]) stands as the solution.
