To find the inverse of the function [tex]\( y = 2x^2 - 4 \)[/tex], we need to follow several steps.
1. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The given equation is:
[tex]\[
y = 2x^2 - 4
\][/tex]
To find the inverse, we swap [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
[tex]\[
x = 2y^2 - 4
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
Now, we need to isolate [tex]\( y \)[/tex]:
- First, add 4 to both sides:
[tex]\[
x + 4 = 2y^2
\][/tex]
- Next, divide both sides by 2:
[tex]\[
\frac{x + 4}{2} = y^2
\][/tex]
- Finally, take the square root of both sides. Remember to consider both the positive and negative roots:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
3. Write the inverse function:
The inverse function can be expressed as:
[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]
From the given choices, this corresponds to:
[tex]\[
y = \pm \sqrt{\frac{x+4}{2}}
\][/tex]
