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.