To find the inverse of the function \( y = 2x^2 - 8 \), we need to follow a series of steps. Here is the detailed step-by-step solution:
1. Express \( y \) in terms of \( x \):
Given the function:
[tex]\[
y = 2x^2 - 8
\][/tex]
2. Swap \( x \) and \( y \):
To find the inverse, we interchange \( x \) and \( y \):
[tex]\[
x = 2y^2 - 8
\][/tex]
3. Solve for \( y \):
We need to isolate \( y \) on one side. Start by solving the equation for \( y \):
[tex]\[
x = 2y^2 - 8
\][/tex]
Add 8 to both sides:
[tex]\[
x + 8 = 2y^2
\][/tex]
Divide both sides by 2:
[tex]\[
\frac{x + 8}{2} = y^2
\][/tex]
Take the square root of both sides. Remember that taking the square root introduces a plus and minus:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
4. Verify the solution:
Thus, the inverse functions derived from the original function are:
[tex]\[
y = \sqrt{\frac{x + 8}{2}} \quad \text{and} \quad y = -\sqrt{\frac{x + 8}{2}}
\][/tex]
Given these steps and the derived expressions, the correct inverse equations align with the option:
[tex]\[
y= \pm \sqrt{\frac{x}{2} + 8}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{y= \pm \sqrt{\frac{x}{2} + 8}}
\][/tex]
