To find the inverse of the function [tex]\( y = 2x^2 - 4 \)[/tex], we need to follow these steps:
1. Start by interchanging [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the given equation.
2. Solve the resulting equation for [tex]\(y\)[/tex].
Let's begin:
### Step 1: Interchange [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
Given:
[tex]\[ y = 2x^2 - 4 \][/tex]
Interchange [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x = 2y^2 - 4 \][/tex]
### Step 2: Solve for [tex]\(y\)[/tex]
Now we solve the equation [tex]\( x = 2y^2 - 4 \)[/tex] for [tex]\(y\)[/tex].
First, isolate the term containing [tex]\(y\)[/tex]:
[tex]\[ x + 4 = 2y^2 \][/tex]
Divide both sides by 2:
[tex]\[ \frac{x + 4}{2} = y^2 \][/tex]
Take the square root of both sides:
[tex]\[ y = \pm \sqrt{\frac{x + 4}{2}} \][/tex]
Therefore, the inverse of the function [tex]\( y = 2x^2 - 4 \)[/tex] is:
[tex]\[ y = \pm \sqrt{\frac{x + 4}{2}} \][/tex]
This matches one of the given options:
[tex]\[ y = \pm \sqrt{\frac{x + 4}{2}} \][/tex]
Thus, the correct answer is:
[tex]\[ y = \pm \sqrt{\frac{x + 4}{2}} \][/tex]
