Use the function  f(x)=x3−1
with its graph shown below to answer the following question. Two of the following statements are true. Select the two that apply. The function  f and its inverse are reflections over the line  y=x
.

The function  f and its inverse both have minimum and maximum values.

The function  f and its inverse are symmetric around the same point.

The function  f and its inverse are both increasing functions.

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.**

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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).

Use the function f(x) = x^3 − 1 with its graph shown below to answer the following question. Two of the following statements are true. Select the two that apply. The function f and its inverse are reflections over the line y = x. The function f and its inverse both have minimum and maximum values. The function f and its inverse are symmetric around the same point. The function f and its inverse are both increasing functions.

Explore the properties of the cubic function f(x) = x^3 - 1 and its inverse, focusing on reflections over the line y = x and the monotonicity of both functions. Learn how to identify key features of these functions with detailed analysis, suitable for Grades 10-12.

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