To find the inverse of the equation [tex]\( y = 2x^2 - 8 \)[/tex], we need to solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] and then swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Here is a step-by-step solution:
1. Start with the original equation:
[tex]\[ y = 2x^2 - 8 \][/tex]
2. Add 8 to both sides to begin isolating the [tex]\( x^2 \)[/tex] term:
[tex]\[ y + 8 = 2x^2 \][/tex]
3. Divide both sides by 2 to further isolate [tex]\( x^2 \)[/tex]:
[tex]\[ \frac{y + 8}{2} = x^2 \][/tex]
4. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \pm \sqrt{\frac{y + 8}{2}} \][/tex]
5. Interchange [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to write the inverse function:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
After working through these steps, we see that the correct inverse function is:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
Therefore, the correct choice from the given options is:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
The correct answer is the first option.
