Given the function $f(x) = -\frac{1}{2}x^2 + 2x - 4$, we need to find its inverse and restrict the domain such that the inverse is also a function.

Step 1: Find the inverse function.

To find the inverse, we need to express $x$ in terms of $y$. Start by substituting $y$ for $f(x)$:

$y = -\frac{1}{2}x^2 + 2x - 4$

Next, multiply the entire equation by $-2$ to eliminate the fraction:

$-2y = x^2 - 4x + 8$

Then, rearrange the equation to make it easier to solve for $x$:

$x^2 - 4x + (8 + 2y) = 0$

This is a quadratic equation in $x$. Use the quadratic formula to solve for $x$:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(8+2y)}}{2(1)}$

Simplify the expression:

$x = \frac{4 \pm \sqrt{16 - 32 - 8y}}{2}$

$x = \frac{4 \pm \sqrt{-8y - 16}}{2}$

$x = \frac{4 \pm 2\sqrt{-2y - 4}}{2}$

$x = 2 \pm \sqrt{-2y - 4}$

Finally, switch $x$ and $y$ to express the inverse function:

$y = 2 \pm \sqrt{-2x - 4}$

Thus, the inverse function $f^{-1}(x)$ is:

$f^{-1}(x) = 2 \pm \sqrt{-2x - 4}$

Step 2: Restrict the domain of $f(x)$.

For the inverse to be a function, the square root expression must be non-negative and only one branch of the square root should be taken. This means we must ensure that $\sqrt{-2x - 4}$ is defined and the domain is restricted so that the quadratic function is either strictly increasing or decreasing.

The expression inside the square root, $-2x - 4$, must be non-negative:

$-2x - 4 \geq 0$

Solving for $x$:

$-2x \geq 4 \quad \Rightarrow \quad x \leq -2$

Thus, the domain of $f(x)$ is restricted to $x \leq -2$.

Step 3: Identify the possible restrictions.

To make $f^{-1}(x)$ a function, choose either the $+$ or $-$ branch for the square root. We have two cases:

1. Case 1: If $f^{-1}(x) = 2 + \sqrt{-2x - 4}$, restrict the domain of $f(x)$ to $(-\infty, -2]$.
2. Case 2: If $f^{-1}(x) = 2 - \sqrt{-2x - 4}$, restrict the domain of $f(x)$ to $[2, \infty)$.

Final Answer:

To ensure that $f^{-1}(x)$ is a function, the function $f(x)$ can have one of the following domain restrictions:

1. $f(x)$ has domain $(-\infty, -2]$, with inverse $f^{-1}(x) = 2 + \sqrt{-2x - 4}$.
2. $f(x)$ has domain $[2, \infty)$, with inverse $f^{-1}(x) = 2 - \sqrt{-2x - 4}$.

Would you like a deeper explanation of any step, or do you have any questions?

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Related Questions:

1. How does restricting the domain of a quadratic function affect its graph?
2. What is the significance of the discriminant in the quadratic formula when finding inverses?
3. How do you determine whether to use the positive or negative branch in the inverse function?
4. How can we verify that a function is one-to-one before finding its inverse?
5. What are other methods for finding the inverse of a quadratic function?

Tip: Always check if the original function is one-to-one over the restricted domain to ensure its inverse exists as a function.