To find the inverse of the function \( y = 2x^2 - 4 \), follow these steps:
1. Start with the given function:
[tex]\[
y = 2x^2 - 4
\][/tex]
2. Swap \(y\) and \(x\): To find the inverse, we interchange the roles of \(x\) and \(y\).
[tex]\[
x = 2y^2 - 4
\][/tex]
3. Solve for \(y\): We need to isolate \(y\) in terms of \(x\).
- First, add 4 to both sides to start isolating \(y^2\):
[tex]\[
x + 4 = 2y^2
\][/tex]
- Next, divide both sides by 2 to further isolate \(y^2\):
[tex]\[
\frac{x + 4}{2} = y^2
\][/tex]
- Finally, take the square root of both sides to solve for \(y\):
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
4. State the inverse functions:
The resulting expressions represent the inverse functions of the given original function. Therefore, we have:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
And equivalently, since taking the square root of a fraction is the same as taking the square root of the numerator and denominator separately:
[tex]\[
y = \pm \frac{\sqrt{x + 4}}{2}
\][/tex]
Thus, the inverse functions of the original function \( y = 2x^2 - 4 \) are:
[tex]\[
y = \pm \sqrt{\frac{x + 4}{2}}
\][/tex]
or equivalently:
[tex]\[
y = \pm \frac{\sqrt{x + 4}}{2}
\][/tex]
So, the correct answer is:
[tex]\[ y = \pm \sqrt{\frac{x+4}{2}} \][/tex]
or
[tex]\[ y = \pm \frac{\sqrt{x+4}}{2} \][/tex]
