To find the inverse of the function $f(x) = x^2 + 8x - 2$, we need to express $x$ in terms of $y$ where $y = f(x)$. Let's follow the steps to determine the inverse.

Step-by-step Solution:

1. Rewrite the function using $y$:

   $y = x^2 + 8x - 2$

2. Solve for $x$ in terms of $y$:

   Rearrange the equation:

   $x^2 + 8x = y + 2$

3. Complete the square:

   To complete the square, we take the coefficient of $x$, which is 8, divide it by 2 to get 4, and then square it to get 16. Add and subtract 16 on the left side:

   $x^2 + 8x + 16 = y + 2 + 16$

   $(x + 4)^2 = y + 18$

4. Solve for $x$:

   Take the square root of both sides:

   $x + 4 = \pm \sqrt{y + 18}$

   $x = -4 \pm \sqrt{y + 18}$

5. Determine the correct branch for the inverse:

   Since the original function $f(x) = x^2 + 8x - 2$ is a parabola that opens upwards, it is not one-to-one over all real numbers. We need to restrict the domain to ensure that the function is invertible. Typically, we choose either the left side (decreasing part) or the right side (increasing part).

   To make the inverse a function, we choose the increasing part, so:

   $x = -4 + \sqrt{y + 18}$

6. Express the inverse function:

   Substitute $y$ back with $x$ to get the inverse function:

   $f^{-1}(x) = -4 + \sqrt{x + 18}$

Conclusion:

The inverse function of $f(x) = x^2 + 8x - 2$ is:

$f^{-1}(x) = -4 + \sqrt{x + 18}, \quad x \geq -18$

This inverse is valid for $x \geq -18$ because the square root function requires non-negative input.

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Would you like more details or have any questions about this solution?

Here are 5 related questions to consider:

1. How do you determine the domain and range of the inverse function?
2. What is the graph of the inverse function $f^{-1}(x) = -4 + \sqrt{x + 18}$?
3. How do you verify if two functions are inverses of each other?
4. Can you find the inverse for other quadratic functions, like $f(x) = x^2 - 6x + 5$?
5. How would the inverse function change if the parabola opened downward?

Tip: To find the inverse of a function, first check if it is one-to-one. If it isn't, consider restricting its domain to ensure the inverse is well-defined.