To find the inverse of the function [tex]\( y = 2x^2 - 4 \)[/tex], we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex].
1. Start with the given function:
[tex]\[
y = 2x^2 - 4
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[
y + 4 = 2x^2
\][/tex]
3. Isolate [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{y + 4}{2}
\][/tex]
4. Take the square root of both sides: Note that taking the square root introduces both the positive and negative roots.
[tex]\[
x = \pm \sqrt{\frac{y + 4}{2}}
\][/tex]
Therefore, the inverse functions are:
[tex]\[
x = \sqrt{\frac{y + 4}{2}} \quad \text{and} \quad x = -\sqrt{\frac{y + 4}{2}}
\][/tex]
In other words, the inverse of the function [tex]\( y = 2x^2 - 4 \)[/tex] is:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
Hence, the correct choice from the given options is:
[tex]\[
y = \pm \sqrt{\frac{x+4}{2}}
\][/tex]
