Inverse of f(X)=-x^2+4x-4 for x is less than or equal to 2

To find the inverse of the function \( f(x) = -x^2 + 4x - 4 \) for \( x \leq 2 \), we need to follow these steps:

### 1. **Express the function as \( y = f(x) \):**
\[
y = -x^2 + 4x - 4
\]

### 2. **Solve for \( x \) in terms of \( y \):**
We need to solve the quadratic equation for \( x \) in terms of \( y \).

\[
-x^2 + 4x - 4 = y
\]

To make it easier to solve, multiply both sides by -1:

\[
x^2 - 4x + 4 = -y
\]

This simplifies to:

\[
x^2 - 4x + (4 + y) = 0
\]

Now, this is a standard quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -4 \), and \( c = 4 + y \).

### 3. **Use the quadratic formula to solve for \( x \):**
The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting \( a = 1 \), \( b = -4 \), and \( c = 4 + y \):

\[
x = \frac{4 \pm \sqrt{16 - 4(1)(4 + y)}}{2(1)}
\]

Simplify the expression under the square root:

\[
x = \frac{4 \pm \sqrt{16 - 16 - 4y}}{2}
\]

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

\[
x = \frac{4 \pm 2i\sqrt{y}}{2}
\]

However, because we are dealing with real values of \( y \) and we need \( x \) to be real (since \( y = f(x) \) and we're only considering \( x \leq 2 \)), there must be an issue with the sign or approach.

Let's rewrite the equation:

### 4. **Refactor to avoid complex values:**

Rewriting the original function in a completed square form:

\[
f(x) = -(x^2 - 4x + 4)
\]
\[
f(x) = -(x - 2)^2
\]

So, we have:

\[
y = -(x - 2)^2
\]

\[
-(x - 2)^2 = y
\]

Solving for \( x \):

\[
(x - 2)^2 = -y
\]

Taking the square root on both sides:

\[
x - 2 = \pm \sqrt{-y}
\]

Since \( \sqrt{-y} \) must be a non-negative value and the function is defined for real values, we take the negative square root due to the restriction \( x \leq 2 \):

\[
x - 2 = -\sqrt{-y}
\]

Finally, solve for \( x \):

\[
x = 2 - \sqrt{-y}
\]

### 5. **Express the inverse function:**
Thus, the inverse function \( f^{-1}(y) \) is:

\[
f^{-1}(y) = 2 - \sqrt{-y}, \text{ for } y \leq 0
\]

### Summary:

The inverse function is:

\[
f^{-1}(y) = 2 - \sqrt{-y} \quad \text{for} \quad y \leq 0
\]

---

Would you like more details or have any questions? Here are some related questions:

1. How can you verify if two functions are inverses of each other?
2. What is the domain and range of the original function \( f(x) \)?
3. How does the restriction \( x \leq 2 \) affect the inverse function?
4. What is the graphical interpretation of finding an inverse?
5. How would you find the inverse if the function was defined for \( x > 2 \)?

**Tip:** Always ensure the domain of the original function aligns with the range of the inverse function.

Discover how to find the inverse of the function f(x) = -x^2 + 4x - 4 for x ≤ 2 with detailed steps and explanations. This problem involves inverse functions and quadratic equations, suitable for high school students in Grades 10-12.

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