To determine the nature of the function [tex]\( f(x) = x^3 - x^2 - 9x + 9 \)[/tex], we need to analyze its properties step by step.
### Step 1: Check if it's a function
A function is a relation where each input (x-value) has a unique output (y-value). The given expression [tex]\( f(x) = x^3 - x^2 - 9x + 9 \)[/tex] is a polynomial in [tex]\( x \)[/tex], and polynomials are always functions because for every input [tex]\( x \)[/tex], there is a unique output. Thus, statement A is incorrect.
### Step 2: The vertical line test
The vertical line test helps us determine if a relation is a function by ensuring any vertical line drawn on the graph of the equation intersects the graph at most once. Since [tex]\( f(x) \)[/tex] is a polynomial, it will pass the vertical line test by definition. Therefore, statement B is incorrect.
### Step 3: Determine whether it is one-to-one or many-to-one
To determine if the function is one-to-one or many-to-one, we need to examine its derivative.
1. Compute the derivative of [tex]\( f(x) \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx}(x^3 - x^2 - 9x + 9) \][/tex]
[tex]\[ f'(x) = 3x^2 - 2x - 9 \][/tex]
2. Find critical points by setting the derivative equal to zero:
[tex]\[ 3x^2 - 2x - 9 = 0 \][/tex]
Solve this quadratic equation for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4(3)(-9)}}{2(3)} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 108}}{6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{112}}{6} \][/tex]
[tex]\[ x = \frac{2 \pm 4\sqrt{7}}{6} \][/tex]
[tex]\[ x = \frac{1 \pm 2\sqrt{7}}{3} \][/tex]
The critical points are:
[tex]\[ x_1 = \frac{1 + 2\sqrt{7}}{3}, \quad x_2 = \frac{1 - 2\sqrt{7}}{3} \][/tex]
3. Examine the behavior of [tex]\( f'(x) \)[/tex] around these critical points:
- The function changes its slope (from increasing to decreasing or vice versa) at the critical points.
- This behavior indicates the presence of local minima or maxima.
Since the function has multiple critical points where [tex]\( f'(x) \)[/tex] changes sign, it implies that the function can attain the same value at different points. This shows [tex]\( f(x) \)[/tex] is not one-to-one. Hence, statement D is incorrect.
### Step 4: Final determination
Given the polynomial has critical points showing changes in concavity, the function can map different [tex]\( x \)[/tex] values to the same [tex]\( y \)[/tex] value. Therefore, [tex]\( f(x) = x^3 - x^2 - 9x + 9 \)[/tex] is many-to-one, and statement C is correct.
### Conclusion
The statement that best describes the function [tex]\( f(x) = x^3 - x^2 - 9x + 9 \)[/tex] is:
C. It is a many-to-one function.
