To find the inverse of the function [tex]\( y = 2x^2 - 8 \)[/tex], we can follow these steps:
1. Switch [tex]\(x\)[/tex] and [tex]\(y\)[/tex]: Start by replacing [tex]\(y\)[/tex] with [tex]\(x\)[/tex] and [tex]\(x\)[/tex] with [tex]\(y\)[/tex] in the equation. This gives us:
[tex]\[
x = 2y^2 - 8
\][/tex]
2. Solve for [tex]\(y\)[/tex]:
- First, isolate the term with [tex]\(y\)[/tex] on one side. Add 8 to both sides of the equation:
[tex]\[
x + 8 = 2y^2
\][/tex]
- Next, divide both sides by 2 to get:
[tex]\[
\frac{x + 8}{2} = y^2
\][/tex]
- Now, take the square root of both sides. Remember that taking the square root introduces both the positive and negative roots:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
Thus, the inverse function of [tex]\( y = 2x^2 - 8 \)[/tex] is:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
Among the given choices, this corresponds to the first option:
[tex]\[
y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
So, the correct answer is:
[tex]\[
1 \quad y = \pm \sqrt{\frac{x + 8}{2}}
\][/tex]
