Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Explore thousands of questions and answers from knowledgeable experts in various fields on our Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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.
### 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.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
