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

### Step 1: Express \( y \) in terms of \( x \)
\[
y = x^2 - 4x
\]

### Step 2: Solve for \( x \) in terms of \( y \)
We rewrite the equation and complete the square to solve for \( x \).

\[
y = x^2 - 4x
\]
To complete the square:
\[
y = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4
\]
Now, the equation is:
\[
y + 4 = (x - 2)^2
\]

### Step 3: Take the square root of both sides
\[
\sqrt{y + 4} = x - 2
\]

### Step 4: Solve for \( x \)
\[
x = 2 \pm \sqrt{y + 4}
\]

### Step 5: Choose the appropriate branch
Because the function \( f(x) = x^2 - 4x \) is not one-to-one without restrictions, we must restrict the domain to make it invertible. For example, if we restrict \( f(x) \) to \( x \geq 2 \), we take the positive branch:
\[
x = 2 + \sqrt{y + 4}
\]

### Final Answer:
The inverse function is:
\[
f^{-1}(y) = 2 + \sqrt{y + 4}
\]
(for \( x \geq 2 \)).

Would you like more details on any step or have any questions?

Here are 5 related questions:
1. How do you complete the square in a general quadratic expression?
2. How does the domain restriction affect the inverse function?
3. What are the conditions for a function to have an inverse?
4. How do you verify if a function and its inverse are correct?
5. What is the geometric interpretation of an inverse function?

**Tip:** When finding inverses of quadratic functions, always consider restricting the domain to make the function one-to-one!

Learn how to find the inverse of the quadratic function f(x) = x² - 4x through completing the square and domain restriction. This detailed explanation includes steps like solving for x and applying the inverse function theorem, suitable for students in Grades 9-12.

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