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Let's go through each part of this problem step-by-step.

### Part (a): Sketch the Graph of \( f \)
The function given is:
\[
f(x) = x^3 - 2
\]

To sketch the graph:
1. **Basic Shape**: The function \( x^3 \) is a cubic function with an inflection point at \( x = 0 \). It has the general shape of an “S,” with values decreasing for negative \( x \) and increasing for positive \( x \).
2. **Vertical Shift**: The subtraction of 2 shifts the entire graph down by 2 units.
3. **Intercepts**:
   - **y-intercept**: When \( x = 0 \), \( f(0) = 0^3 - 2 = -2 \). So, the y-intercept is at \( (0, -2) \).
   - **x-intercept**: Set \( f(x) = 0 \):
     \[
     x^3 - 2 = 0 \Rightarrow x^3 = 2 \Rightarrow x = \sqrt[3]{2}
     \]
     The x-intercept is approximately at \( x \approx 1.26 \).

The resulting graph will have the typical cubic shape but shifted downward by 2 units.

### Part (b): Sketch the Graph of \( f^{-1} \)
The inverse function graph can be obtained by reflecting the graph of \( f(x) \) over the line \( y = x \).
1. The y-intercept of \( f(x) \) becomes the x-intercept of \( f^{-1}(x) \).
2. The x-intercept of \( f(x) \), approximately \( (\sqrt[3]{2}, 0) \), becomes a point on the y-axis for \( f^{-1}(x) \), namely \( (0, \sqrt[3]{2}) \).

### Part (c): Find the Formula for \( f^{-1}(x) \)
To find the inverse function \( f^{-1}(x) \):
1. Start with \( y = x^3 - 2 \).
2. Swap \( x \) and \( y \) to reflect finding the inverse:
   \[
   x = y^3 - 2
   \]
3. Solve for \( y \):
   \[
   y^3 = x + 2
   \]
   \[
   y = \sqrt[3]{x + 2}
   \]
Thus, the formula for the inverse function is:
\[
f^{-1}(x) = \sqrt[3]{x + 2}
\]

### Summary
- **(a)**: The graph of \( f(x) = x^3 - 2 \) is a cubic function shifted downward by 2.
- **(b)**: To sketch \( f^{-1}(x) \), reflect \( f(x) \) over the line \( y = x \).
- **(c)**: The inverse function is \( f^{-1}(x) = \sqrt[3]{x + 2} \).

Would you like any further details on the solution or help with graphing these functions? 

#### Related Questions
1. What are the steps for finding the inverse of a function algebraically?
2. How does shifting a function affect its graph?
3. What is the significance of reflecting a function over the line \( y = x \) for finding its inverse?
4. How can you determine if a function has an inverse by looking at its graph?
5. What are some common transformations of cubic functions?

#### Tip
When sketching the inverse of a function, always remember that the graph of the inverse is the reflection of the original function over the line \( y = x \). This reflection swaps the x- and y-values of points on the original function.

For the function f(x) = x^3 - 2, (a) sketch the graph of f, (b) use the graph of f to sketch the graph of f^(-1), (c) enter the correct formula for f^(-1)(x).

Learn how to find the inverse of the cubic function f(x) = x^3 - 2. This includes sketching the graph of the function and its inverse, understanding graph transformations, and applying the reflection rule over y = x. Suitable for students in Grades 10-12.

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