Find the inverse. Check your answer algebraically and graphically.

*f*(*x*) = *x*2 − 8*x* − 8,    *x* ≤ −4

To find the inverse of the function \( f(x) = x^2 - 8x - 8 \) with the condition \( x \leq -4 \), let's break it down step by step. 

### Step 1: Rewrite the function and find the inverse.

Given:  
\[
f(x) = x^2 - 8x - 8
\]

1. We need to solve for \( x \) in terms of \( y \). So, let \( f(x) = y \):
   \[
   y = x^2 - 8x - 8
   \]

2. **Complete the square** to rewrite the quadratic expression:

   \[
   y = (x^2 - 8x) - 8
   \]
   Now, to complete the square:
   \[
   x^2 - 8x = (x - 4)^2 - 16
   \]
   So the equation becomes:
   \[
   y = (x - 4)^2 - 16 - 8 = (x - 4)^2 - 24
   \]

3. Now, express \( x \) in terms of \( y \):

   \[
   y + 24 = (x - 4)^2
   \]
   Take the square root of both sides:
   \[
   \sqrt{y + 24} = |x - 4|
   \]
   Since \( x \leq -4 \), we choose the negative branch of the square root:
   \[
   \sqrt{y + 24} = -(x - 4)
   \]
   Simplifying for \( x \):
   \[
   x = 4 - \sqrt{y + 24}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = 4 - \sqrt{x + 24}
\]

### Step 2: Verify algebraically

To check if the inverse is correct, we need to confirm that applying \( f \) and then \( f^{-1} \) results in the original input.

1. Start with \( f(f^{-1}(x)) \):

\[
f(f^{-1}(x)) = f(4 - \sqrt{x + 24})
\]
Substitute \( 4 - \sqrt{x + 24} \) into the original function \( f(x) = x^2 - 8x - 8 \):

\[
f(4 - \sqrt{x + 24}) = (4 - \sqrt{x + 24})^2 - 8(4 - \sqrt{x + 24}) - 8
\]
Simplifying:
\[
= (16 - 8\sqrt{x + 24} + (x + 24)) - 32 + 8\sqrt{x + 24} - 8
\]
\[
= 16 - 32 - 8 + (x + 24) = x
\]

So, \( f(f^{-1}(x)) = x \), confirming that the inverse is correct.

### Step 3: Graphical Verification

Graph both the function \( f(x) = x^2 - 8x - 8 \) (restricted to \( x \leq -4 \)) and its inverse \( f^{-1}(x) = 4 - \sqrt{x + 24} \).

- The original function is a parabola opening upwards, and since \( x \leq -4 \), we only take the left half of the parabola.
- The inverse function is the reflection of the original function across the line \( y = x \).

Both graphs should reflect each other across the line \( y = x \), confirming that they are indeed inverses.

Would you like a graphical representation of this? Let me know if you'd like further clarification or more details.

### 5 Questions to Consider:
1. How is completing the square helpful in finding the inverse of a quadratic function?
2. Why do we take the negative square root branch when finding the inverse in this case?
3. What happens graphically when you reflect a function and its inverse across \( y = x \)?
4. What role does the domain restriction \( x \leq -4 \) play in defining the inverse?
5. How does the verification process show that two functions are inverses?

**Tip:** When finding inverses of quadratic functions, always check if there are domain restrictions to ensure the inverse exists!

Find the inverse. Check your answer algebraically and graphically.

*f*(*x*) = *x*^2 − 8*x* − 8,    *x* ≤ −4

This page provides a step-by-step solution to finding the inverse of the quadratic function f(x) = x^2 - 8x - 8 with the restriction x ≤ -4. The solution involves completing the square and verifying the result algebraically and graphically. Suitable for Grades 10-12 students learning about inverse functions and quadratic equations.

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