To find the inverse function of $y = x^2 - 6x + 8$, follow these steps:

Step 1: Express $y$ as a function of $x$.

The given function is: $y = x^2 - 6x + 8$

Step 2: Replace $y$ with $x$ and $x$ with $y$.

To find the inverse, swap $x$ and $y$: $x = y^2 - 6y + 8$

Step 3: Solve for $y$.

Now, solve this quadratic equation for $y$. First, rewrite the equation as: $y^2 - 6y + (8 - x) = 0$

This is a standard quadratic equation of the form $ay^2 + by + c = 0$, where $a = 1$, $b = -6$, and $c = 8 - x$.

Step 4: Use the quadratic formula.

The quadratic formula is: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute the values of $a$, $b$, and $c$ into the formula: $y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(8 - x)}}{2(1)}$ $y = \frac{6 \pm \sqrt{36 - 32 + 4x}}{2}$ $y = \frac{6 \pm \sqrt{4 + 4x}}{2}$ $y = \frac{6 \pm \sqrt{4(1 + x)}}{2}$ $y = \frac{6 \pm 2\sqrt{1 + x}}{2}$ $y = 3 \pm \sqrt{1 + x}$

Step 5: Consider the correct branch.

The original function $y = x^2 - 6x + 8$ is a parabola that opens upwards. The inverse function can have two branches, corresponding to the $\pm$ sign. To maintain the function nature of the inverse, we must choose one branch based on the domain and range. If the domain of the original function is restricted to where the function is increasing (after the vertex), then the inverse is: $y = 3 + \sqrt{1 + x}$

Final Answer:

The inverse function, considering the positive branch (assuming we are dealing with the domain where $x \geq 3$), is: $f^{-1}(x) = 3 + \sqrt{1 + x}$

If you want to use the negative branch (where $x \leq 3$), the inverse would be: $f^{-1}(x) = 3 - \sqrt{1 + x}$

Additional Questions:

1. What are the domain and range of the original function $y = x^2 - 6x + 8$?
2. How do you determine which branch of the inverse function to choose?
3. Can the function have an inverse over its entire domain?
4. How would the graph of the original function and its inverse look like?
5. What is the importance of the vertex in finding the inverse of a quadratic function?

Tip: When finding the inverse of a quadratic function, the domain must often be restricted to ensure that the inverse is a function.