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!