To find the inverse of the function [tex]\( y = 2x^2 + 2 \)[/tex], we need to follow these steps.
### Step-by-Step Solution:
1. Write the function:
[tex]\[
y = 2x^2 + 2
\][/tex]
2. To find the inverse, swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 2y^2 + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
First, subtract 2 from both sides:
[tex]\[
x - 2 = 2y^2
\][/tex]
Then divide both sides by 2:
[tex]\[
\frac{x - 2}{2} = y^2
\][/tex]
4. Take the square root of both sides:
[tex]\[
y = \pm \sqrt{\frac{x - 2}{2}}
\][/tex]
5. Simplify the expression:
The two solutions are:
[tex]\[
y = \sqrt{\frac{x - 2}{2}} \quad \text{and} \quad y = -\sqrt{\frac{x - 2}{2}}
\][/tex]
Hence, the inverse functions are:
[tex]\[
y = \sqrt{\frac{x - 2}{2}} \quad \text{and} \quad y = -\sqrt{\frac{x - 2}{2}}
\][/tex]
These solutions correspond to:
[tex]\[
\pm \sqrt{\frac{x - 2}{2}}
\][/tex]
Thus, the correct choice from the given options is:
[tex]\[
y = \pm \sqrt{\frac{x - 2}{2}}
\][/tex]
However, it appears that there is a slight error in the provided options. Correcting the inverse function form, we get:
[tex]\[
y = \pm \sqrt{\frac{x - 2}{2}}
\][/tex]
So the actual correct form which wasn't perfectly matched in the given options is:
[tex]\[
\pm \sqrt{\frac{x - 2}{2}}
\][/tex]
