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Using the horizontal-line test, determine whether the function is one-to-one.

f(x) = x3 - 3x + 2

To determine whether the function \( f(x) = x^3 - 3x + 2 \) is one-to-one using the **horizontal-line test**, we must analyze the graph.

### Horizontal-Line Test Explanation:
A function is **one-to-one** if and only if no horizontal line intersects the graph of the function at more than one point. This ensures that for every output value \( y \), there is exactly one corresponding input value \( x \), meaning the function has an inverse.

### Observation from the Graph:
Looking at the graph you provided, the function crosses some horizontal lines at multiple points. For instance, horizontal lines between \( y = -4 \) and \( y = 4 \) intersect the graph at more than one point. This indicates that the function is **not one-to-one**, because it fails the horizontal-line test.

### Conclusion:
Since the graph fails the horizontal-line test, the function \( f(x) = x^3 - 3x + 2 \) is **not** one-to-one.

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:
1. What is the **inverse** of a one-to-one function?
2. How do you perform the **vertical-line test** and what does it determine?
3. Can you use calculus to determine where a function is **increasing or decreasing**?
4. What are the **critical points** of the function \( f(x) = x^3 - 3x + 2 \)?
5. How does the **derivative** of a function relate to its monotonicity?

**Tip:** When analyzing graphs, the derivative \( f'(x) \) helps identify intervals where the function is increasing or decreasing, which can aid in determining if the function is one-to-one.

Using the horizontal-line test, determine whether the function is one-to-one.
f(x) = x^3 - 3x + 2

This problem explores the one-to-one nature of the function f(x) = x^3 - 3x + 2 using the horizontal-line test. The graph indicates that the function is not one-to-one as horizontal lines intersect the graph at multiple points. Suitable for Grades 10-12 students learning about graphing and function behavior.

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