To find the inverse of the function [tex]\( y = 2x^2 + 2 \)[/tex], let's proceed step-by-step.
1. Rewrite the function with [tex]\( y \)[/tex] as the dependent variable:
[tex]\[
y = 2x^2 + 2
\][/tex]
2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = 2y^2 + 2
\][/tex]
3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x - 2 = 2y^2
\][/tex]
[tex]\[
\frac{x - 2}{2} = y^2
\][/tex]
4. Take the square root of both sides:
[tex]\[
y = \pm \sqrt{\frac{x - 2}{2}}
\][/tex]
5. Express [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in terms of [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[
y = \pm \frac{\sqrt{2x - 4}}{2}
\][/tex]
Therefore, the inverse of [tex]\( y = 2x^2 + 2 \)[/tex] is:
[tex]\[
y = -\frac{\sqrt{2x - 4}}{2} \quad \text{and} \quad y = \frac{\sqrt{2x - 4}}{2}
\][/tex]
The inverse functions are:
[tex]\[
f^{-1}(x) = -\frac{\sqrt{2x - 4}}{2} \quad \text{and} \quad f^{-1}(x) = \frac{\sqrt{2x - 4}}{2}
\][/tex]
These represent the two branches of the inverse function, reflecting the fact that [tex]\( y = 2x^2 + 2 \)[/tex] is a parabola which is symmetric about the y-axis.
