Answer :
To determine which statement best describes the function \( f(x) = x^3 - x^2 - 9x + 9 \), we need to analyze its properties, particularly focusing on whether the function is one-to-one or many-to-one.
### Step-by-Step Solution
1. Definition of One-to-One Function:
- A function is one-to-one if every horizontal line intersects the graph of the function at most once. This means that for every value of \( y \), there is at most one corresponding value of \( x \).
2. Consider the Derivative:
- To determine if \( f(x) \) is one-to-one, we should examine its first derivative, \( f'(x) \). The first derivative tells us about the slope of the function.
- Calculate the first derivative of \( f(x) \):
[tex]\[ f'(x) = 3x^2 - 2x - 9 \][/tex]
3. Critical Points:
- To find the critical points, set the first derivative equal to zero:
[tex]\[ 3x^2 - 2x - 9 = 0 \][/tex]
4. Roots of the Derivative:
- Solve the quadratic equation \( 3x^2 - 2x - 9 = 0 \). The roots of this equation are the values of \( x \) where the slope of \( f(x) \) is zero (where the graph has horizontal tangents).
5. Nature of Roots:
- Determine if the roots are real and distinct. If the roots are real and distinct, the function's derivative does not indicate repeated tangents, which suggests that it is one-to-one.
6. Conclusion:
- After solving, we find that \( f(x) \) does not have repeated real roots in its derivative, indicating that it does not pass multiple critical points with the same slope. With unique critical points, the overall behavior suggests that the function is one-to-one.
Hence, the function \( f(x) = x^3 - x^2 - 9x + 9 \) is best described by:
D. It is a one-to-one function.
### Step-by-Step Solution
1. Definition of One-to-One Function:
- A function is one-to-one if every horizontal line intersects the graph of the function at most once. This means that for every value of \( y \), there is at most one corresponding value of \( x \).
2. Consider the Derivative:
- To determine if \( f(x) \) is one-to-one, we should examine its first derivative, \( f'(x) \). The first derivative tells us about the slope of the function.
- Calculate the first derivative of \( f(x) \):
[tex]\[ f'(x) = 3x^2 - 2x - 9 \][/tex]
3. Critical Points:
- To find the critical points, set the first derivative equal to zero:
[tex]\[ 3x^2 - 2x - 9 = 0 \][/tex]
4. Roots of the Derivative:
- Solve the quadratic equation \( 3x^2 - 2x - 9 = 0 \). The roots of this equation are the values of \( x \) where the slope of \( f(x) \) is zero (where the graph has horizontal tangents).
5. Nature of Roots:
- Determine if the roots are real and distinct. If the roots are real and distinct, the function's derivative does not indicate repeated tangents, which suggests that it is one-to-one.
6. Conclusion:
- After solving, we find that \( f(x) \) does not have repeated real roots in its derivative, indicating that it does not pass multiple critical points with the same slope. With unique critical points, the overall behavior suggests that the function is one-to-one.
Hence, the function \( f(x) = x^3 - x^2 - 9x + 9 \) is best described by:
D. It is a one-to-one function.
