How to Find the Inverse of the Function y = 2x² + 2?

To find the inverse of the function given by y = 2x² + 2, we need to follow these steps:

1. First, replace y with x:
2. x = 2y² + 2
3. Now, isolate y²:

Subtract 2 from both sides:

x – 2 = 2y²

Now, divide by 2:

y² = (x – 2) / 2

y = ±√((x – 2) / 2)

However, we need to determine which part of the square root to take. The original function y = 2x² + 2 is a parabola that opens upwards and has its vertex at (0, 2). To keep the inverse a function, we only consider the positive square root:

y = √((x – 2) / 2)

So, the inverse function is:

f^-1(x) = √((x – 2) / 2)

This inverse will only be valid for values of x that are greater than or equal to 2, as those are the values that correspond to y in the original function.