Answer :
To determine which statement best describes the function \( f(x) = 2x^3 + 2x^2 - x \), let's analyze its characteristics step by step.
### Step 1: Understanding the Function
First, observe that \( f(x) = 2x^3 + 2x^2 - x \) is a polynomial. Polynomials are always functions because they pass the vertical line test.
### Step 2: Vertical Line Test
The vertical line test is used to determine if an equation is a function: if any vertical line intersects the graph of the equation in no more than one point, then it is a function. Since \( f(x) \) is a polynomial, it is inherently a function because polynomials pass the vertical line test. Thus, Option D is incorrect.
### Step 3: Determine One-to-One or Many-to-One
A function is one-to-one if every element of the range corresponds to exactly one element of the domain. A function is many-to-one if at least two different elements of the domain map to the same element of the range.
#### Derivative and Critical Points
To investigate whether our function is one-to-one or many-to-one, we can check its derivative to find critical points and analyze monotonicity:
[tex]\[ f'(x) = \frac{d}{dx}(2x^3 + 2x^2 - x) = 6x^2 + 4x - 1 \][/tex]
Set the derivative equal to zero to find critical points:
[tex]\[ 6x^2 + 4x - 1 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{-4 \pm \sqrt{16 + 24}}{12} = \frac{-4 \pm \sqrt{40}}{12} = \frac{-4 \pm 2\sqrt{10}}{12} = \frac{-2 \pm \sqrt{10}}{6} = \frac{-1 \pm \sqrt{10}/3}{3} \][/tex]
We have two distinct critical points. These indicate changes in the direction of the function, which suggests that the function is not strictly increasing or decreasing. Therefore, there are intervals where the function is increasing and intervals where it is decreasing.
Due to this behavior, it is likely that for some y-values in the range of \( f(x) \), there exist at least two different x-values corresponding to them.
### Conclusion
Given that the function is a polynomial of degree 3 and polynomials of odd degree typically map multiple x-values to the same y-value, it is clear that the function \( f(x) = 2x^3 + 2x^2 - x \) is a many-to-one function.
Thus, the best statement that describes the function is:
C. It is a many-to-one function.
### Step 1: Understanding the Function
First, observe that \( f(x) = 2x^3 + 2x^2 - x \) is a polynomial. Polynomials are always functions because they pass the vertical line test.
### Step 2: Vertical Line Test
The vertical line test is used to determine if an equation is a function: if any vertical line intersects the graph of the equation in no more than one point, then it is a function. Since \( f(x) \) is a polynomial, it is inherently a function because polynomials pass the vertical line test. Thus, Option D is incorrect.
### Step 3: Determine One-to-One or Many-to-One
A function is one-to-one if every element of the range corresponds to exactly one element of the domain. A function is many-to-one if at least two different elements of the domain map to the same element of the range.
#### Derivative and Critical Points
To investigate whether our function is one-to-one or many-to-one, we can check its derivative to find critical points and analyze monotonicity:
[tex]\[ f'(x) = \frac{d}{dx}(2x^3 + 2x^2 - x) = 6x^2 + 4x - 1 \][/tex]
Set the derivative equal to zero to find critical points:
[tex]\[ 6x^2 + 4x - 1 = 0 \][/tex]
Solve the quadratic equation:
[tex]\[ x = \frac{-4 \pm \sqrt{16 + 24}}{12} = \frac{-4 \pm \sqrt{40}}{12} = \frac{-4 \pm 2\sqrt{10}}{12} = \frac{-2 \pm \sqrt{10}}{6} = \frac{-1 \pm \sqrt{10}/3}{3} \][/tex]
We have two distinct critical points. These indicate changes in the direction of the function, which suggests that the function is not strictly increasing or decreasing. Therefore, there are intervals where the function is increasing and intervals where it is decreasing.
Due to this behavior, it is likely that for some y-values in the range of \( f(x) \), there exist at least two different x-values corresponding to them.
### Conclusion
Given that the function is a polynomial of degree 3 and polynomials of odd degree typically map multiple x-values to the same y-value, it is clear that the function \( f(x) = 2x^3 + 2x^2 - x \) is a many-to-one function.
Thus, the best statement that describes the function is:
C. It is a many-to-one function.
