Let's work through each part of this question step-by-step.

Part (a): Find an equation for the inverse of $f(x)$.

Given: $f(x) = x^2 - 2$ To find the inverse, follow these steps:

1. Replace $f(x)$ with $y$: $y = x^2 - 2$

2. Swap $x$ and $y$ to find the inverse: $x = y^2 - 2$

3. Solve for $y$: $y^2 = x + 2$ $y = \pm \sqrt{x + 2}$

Thus, the inverse function is: $f^{-1}(x) = \pm \sqrt{x + 2}$

However, since the original function $f(x) = x^2 - 2$ is not one-to-one (it fails the horizontal line test), its inverse is not a function over the entire domain. We would need to restrict the domain of $f(x)$ to make the inverse a function.

Part (b): Sketch a graph of the inverse of $f(x)$.

The graph of the inverse $f^{-1}(x) = \pm \sqrt{x + 2}$ would look like the reflection of the original parabola over the line $y = x$. Since we have both $+\sqrt{x + 2}$ and $-\sqrt{x + 2}$, the graph will have two branches: one in the positive $y$-axis and one in the negative $y$-axis, starting from $x = -2$.

Part (c): Is the inverse of $f(x)$ a function?

No, the inverse of $f(x)$ is not a function because it has two values (both $+\sqrt{x + 2}$ and $-\sqrt{x + 2}$) for each $x$ in the domain of the inverse. This violates the definition of a function, which requires that each input has exactly one output.

Part (d): Is $f(x)$ a one-to-one function?

No, $f(x) = x^2 - 2$ is not a one-to-one function because it fails the horizontal line test. For example, $f(1) = -1$ and $f(-1) = -1$, so multiple $x$-values map to the same $y$-value, making it non-invertible over its entire domain.

Would you like a more detailed explanation or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How can you restrict the domain of $f(x) = x^2 - 2$ to make it a one-to-one function?
2. What is the range of $f(x) = x^2 - 2$?
3. How do we determine if a function has an inverse function?
4. What does the horizontal line test tell us about functions and their inverses?
5. How would the graph of the inverse $f^{-1}(x)$ change if we restricted $f(x)$ to only positive values of $x$?

Tip: When finding the inverse of a function, always check if the function is one-to-one first. This can save time and clarify whether an inverse function exists over the entire domain.