To find the inverse of the function \( y = 2x^2 - 8 \), we need to follow a sequence of steps to express \( x \) in terms of \( y \) and then solve for \( y \). Here is a detailed, step-by-step solution:
1. Start with the given equation:
[tex]\[
y = 2x^2 - 8
\][/tex]
2. Isolate \( x^2 \):
Add 8 to both sides to get:
[tex]\[
y + 8 = 2x^2
\][/tex]
3. Solve for \( x^2 \):
Divide both sides by 2:
[tex]\[
\frac{y + 8}{2} = x^2
\][/tex]
4. Solve for \( x \):
Take the square root of both sides. Remember that taking the square root introduces both the positive and negative solutions:
[tex]\[
x = \pm \sqrt{\frac{y + 8}{2}}
\][/tex]
5. Express the inverse function:
To express the inverse function \( y \) in terms of \( x \), we interchange \( x \) and \( y \):
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
So, the equation of the inverse function is:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
Hence, the correct inverse equation that matches the given choices is:
[tex]\[ y = \pm \sqrt{\frac{x + 8}{2}} \][/tex]
