To find the inverse of the function $f(x) = x^2 + 4x - 6$, we need to follow these steps:

1. Express the function in a form where we can easily solve for $x$.

First, rewrite the function:

$f(x) = x^2 + 4x - 6$

Complete the square to rewrite the quadratic in a more convenient form:

$x^2 + 4x = (x + 2)^2 - 4$

Now substitute this back into the function:

$f(x) = (x + 2)^2 - 4 - 6 = (x + 2)^2 - 10$

So the function becomes:

$f(x) = (x + 2)^2 - 10$

2. Swap $f(x)$ with $y$ and solve for $x$.

Let $y = (x + 2)^2 - 10$.

Now, solve for $x$ in terms of $y$:

$y + 10 = (x + 2)^2$

Take the square root of both sides:

$x + 2 = \pm \sqrt{y + 10}$

Solve for $x$:

$x = -2 \pm \sqrt{y + 10}$

3. Choose the correct branch.

Since a function must be one-to-one to have an inverse, we need to restrict the domain of the original function to make it one-to-one. We can either consider the domain $x \geq -2$ (the right half of the parabola) or $x \leq -2$ (the left half of the parabola). For the inverse to be a function, we must choose one branch.

Assuming we restrict to $x \geq -2$, we take the positive square root:

$x = -2 + \sqrt{y + 10}$

4. Write the inverse.

Now, replace $y$ with $x$ to express the inverse function:

$f^{-1}(x) = -2 + \sqrt{x + 10}$

Final Answer:

The inverse of $f(x) = x^2 + 4x - 6$ (with the domain restricted to $x \geq -2$) is:

$f^{-1}(x) = -2 + \sqrt{x + 10}$

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Would you like further clarification on this process, or is there any part you need help with?

Related Questions:

1. What is the general method for finding the inverse of a quadratic function?
2. How does completing the square help in solving for the inverse of a quadratic?
3. Why is it important to restrict the domain of a quadratic function when finding its inverse?
4. How can we verify the inverse by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$?
5. What happens if we choose the other branch (the negative square root)?

Tip: Remember that the inverse of a function exists only if the function is one-to-one. For quadratics, this requires restricting the domain.