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].
Here is the step-by-step process to find the inverse:
1. Write the equation:
[tex]\[
y = 2x^2 - 4
\][/tex]
2. Solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
First, add 4 to both sides to isolate the term containing [tex]\( x \)[/tex]:
[tex]\[
y + 4 = 2x^2
\][/tex]
Next, divide both sides by 2 to solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{y + 4}{2}
\][/tex]
Finally, take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{y + 4}{2}}
\][/tex]
3. Express the inverse function:
The inverse function will actually consist of two branches due to the [tex]\( \pm \)[/tex] sign:
[tex]\[
x = \sqrt{\frac{y + 4}{2}} \quad \text{and} \quad x = -\sqrt{\frac{y + 4}{2}}
\][/tex]
We generally express the inverse function with respect to [tex]\( x \)[/tex]:
[tex]\[
f^{-1}(x) = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
So the correct inverse functions are:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
When comparing the given options:
- [tex]\( y = \pm \sqrt{x} + 2 \)[/tex]: Incorrect.
- [tex]\( y = \pm \sqrt{\frac{x + 4}{2}} \)[/tex]: Correct.
- [tex]\( y = \pm \frac{\sqrt{x + 4}}{2} \)[/tex]: Incorrect.
- [tex]\( y = \pm \sqrt{x} - 2 \)[/tex]: Incorrect.
Thus, the correct inverse function for [tex]\( y = 2x^2 - 4 \)[/tex] is:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
