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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?

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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.

Consider the graphed function f(x) = x^2 - 2. 
1. Find an equation for the inverse of f(x).
2. Sketch a graph of the inverse of f(x).
3. Is the inverse of f(x) a function?
4. Is f(x) a one-to-one function?

Learn how to find the inverse of the function f(x) = x^2 - 2, determine if the inverse is a function, and check if f(x) is a one-to-one function. This step-by-step solution covers key concepts in inverse functions and quadratic functions, making it ideal for high school students.

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