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!