What is the INVERSE of f(x) = 2(x - 4)^2 +6

To find the inverse of the function \( f(x) = 2(x - 4)^2 + 6 \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = 2(x - 4)^2 + 6
\]

### Step 2: Swap \( x \) and \( y \)
To find the inverse, we switch \( x \) and \( y \):
\[
x = 2(y - 4)^2 + 6
\]

### Step 3: Solve for \( y \)

#### Subtract 6 from both sides:
\[
x - 6 = 2(y - 4)^2
\]

#### Divide by 2:
\[
\frac{x - 6}{2} = (y - 4)^2
\]

#### Take the square root of both sides:
\[
y - 4 = \pm \sqrt{\frac{x - 6}{2}}
\]

#### Add 4 to both sides:
\[
y = 4 \pm \sqrt{\frac{x - 6}{2}}
\]

### Step 4: Select the positive or negative branch
Since \( f(x) = 2(x - 4)^2 + 6 \) is a **parabola** that opens upwards, it’s a one-to-one function on the **restricted domain where \( x \geq 4 \)**. Therefore, we select the **positive square root** to keep the function and its inverse consistent:

\[
y = 4 + \sqrt{\frac{x - 6}{2}}
\]

### Step 5: Final answer for the inverse
\[
f^{-1}(x) = 4 + \sqrt{\frac{x - 6}{2}}
\]

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Would you like further details on this process or any specific clarifications?

Here are 5 questions to expand on this topic:
1. What are the domain and range of \( f(x) \) and its inverse?
2. How would the inverse change if the parabola opened downwards?
3. How can you verify that the inverse is correct?
4. Why is the inverse a square root function?
5. How would the process change if the function had a different transformation?

**Tip:** To verify an inverse function, check that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

What is the INVERSE of f(x) = 2(x - 4)^2 + 6

Learn how to find the inverse of the quadratic function f(x) = 2(x - 4)^2 + 6. This step-by-step guide explains the key algebraic concepts, such as switching x and y, solving for y, and understanding square root functions. Suitable for high school students in Grades 10-12.

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