Let's analyze the function $f(x) = x^3 - 1$ and its inverse to identify the two correct statements.

1. Reflection over the line $y = x$:

• A function and its inverse are always reflections of each other over the line $y = x$. So, this statement is true.
• True statement: The function $f$ and its inverse are reflections over the line $y = x$.

2. Minimum and maximum values:

• The function $f(x) = x^3 - 1$ is a cubic function, which is monotonic and has no minimum or maximum values (it goes to $-\infty$ as $x \to -\infty$ and $\infty$ as $x \to \infty$).
• Its inverse, therefore, will also not have minimum or maximum values.
• False statement: The function $f$ and its inverse both have minimum and maximum values.

3. Symmetry around the same point:

• The function $f(x) = x^3 - 1$ is neither symmetric around a point nor has any symmetry about a line (except for the reflection over $y = x$ with its inverse). So, this is false.
• False statement: The function $f$ and its inverse are symmetric around the same point.

4. Both functions are increasing:

• The function $f(x) = x^3 - 1$ is an increasing function because its derivative $f'(x) = 3x^2$ is always positive for all $x$. Since the inverse of a strictly increasing function is also strictly increasing, the inverse function will also be increasing.
• True statement: The function $f$ and its inverse are both increasing functions.

Correct Statements:

1. The function $f$ and its inverse are reflections over the line $y = x$.
2. The function $f$ and its inverse are both increasing functions.

---

Would you like more details or have any questions?

Here are some related questions to explore:

1. What is the derivative of the inverse function of $f(x) = x^3 - 1$?
2. How do you compute the inverse of a cubic function like $f(x) = x^3 - 1$?
3. What does it mean for a function to be monotonic?
4. What are the graphical properties of functions and their inverses?
5. How do reflections across the line $y = x$ affect the graph of a function?

Tip: A function's inverse only exists if the function is one-to-one (i.e., it passes the horizontal line test).