Find the inverse of the function y = 2x2 + 2.

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To find the inverse of the function \( y = 2x^2 + 2 \), follow these steps:

### Step 1: Replace \( y \) with \( f(x) \)
The given function is \( y = 2x^2 + 2 \). So, the function can be written as:
\[
f(x) = 2x^2 + 2
\]

### Step 2: Interchange \( x \) and \( y \)
To find the inverse, interchange \( x \) and \( y \) in the equation:
\[
x = 2y^2 + 2
\]

### Step 3: Solve for \( y \)
Now, solve this equation for \( y \):

1. Subtract 2 from both sides:
\[
x - 2 = 2y^2
\]

2. Divide by 2:
\[
\frac{x - 2}{2} = y^2
\]

3. Take the square root of both sides:
\[
y = \pm \sqrt{\frac{x - 2}{2}}
\]

### Step 4: Specify the domain for the inverse
Since the original function is a parabola (which isn't one-to-one), the inverse will have two branches corresponding to \( y = \sqrt{\frac{x - 2}{2}} \) and \( y = -\sqrt{\frac{x - 2}{2}} \).

To ensure a proper inverse, we typically restrict the domain of the original function to make it one-to-one. For example, we could restrict \( x \geq 0 \), meaning we only take the positive square root for the inverse.

Thus, the inverse function is:
\[
f^{-1}(x) = \sqrt{\frac{x - 2}{2}}
\]
if \( x \geq 0 \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the domain and range of an inverse function?
2. What is the effect of restricting the domain on the inverse function?
3. Can every quadratic function have an inverse? Why or why not?
4. How do you verify that a function and its inverse are correct?
5. What is the graph of \( f^{-1}(x) \) for the given function?

### Tip:
Always check if the original function is one-to-one before finding the inverse. If not, you may need to restrict the domain.

Find the inverse of the function y = 2x^2 + 2.

Learn how to find the inverse of the quadratic function y = 2x^2 + 2. This problem covers important concepts like inverse functions, domain restrictions, and solving quadratic equations, making it suitable for students in Grades 10-12.

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