Which statement best describes the function below?
$f(x)=2 x^3+2 x^2-x$
A. It fails the vertical line test.
B. It is a one-to-one function.
C. It is a many-to-one function.
D. It is not a function.
Answer (1)
The function passes the vertical line test, so it is a function.
Calculate the derivative of the function: f ′ ( x ) = 6 x 2 + 4 x − 1 .
Find the roots of the derivative: x = 6 − 2 ± 10 .
Since the derivative has two distinct real roots, the function is many-to-one, so the answer is C .
Explanation
Understanding the Function We are given the function f ( x ) = 2 x 3 + 2 x 2 − x and asked to determine which statement best describes it from the given options.
Analyzing the Options Let's analyze each option:
A. It fails the vertical line test: A function fails the vertical line test if a vertical line intersects the graph of the function at more than one point. Polynomial functions, like the one given, always pass the vertical line test. Thus, this option is incorrect.
B. It is a one-to-one function: A function is one-to-one if it is strictly increasing or strictly decreasing. To check this, we can analyze its derivative.
C. It is a many-to-one function: A function is many-to-one if there exist at least two different x values that map to the same y value.
D. It is not a function: The given expression is a polynomial, and all polynomials are functions. Thus, this option is incorrect.
Finding the Derivative To determine if the function is one-to-one or many-to-one, we can analyze its derivative. If the derivative is always positive or always negative, the function is one-to-one. If the derivative changes sign, the function is many-to-one.
First, let's find the derivative of f ( x ) :
f ′ ( x ) = 6 x 2 + 4 x − 1
Finding the Roots of the Derivative Now, let's find the roots of the derivative by setting f ′ ( x ) = 0 :
6 x 2 + 4 x − 1 = 0 We can use the quadratic formula to find the roots: x = 2 a − b ± b 2 − 4 a c = 2 ( 6 ) − 4 ± 4 2 − 4 ( 6 ) ( − 1 ) = 12 − 4 ± 16 + 24 = 12 − 4 ± 40 = 12 − 4 ± 2 10 = 6 − 2 ± 10 So the roots are x 1 = 6 − 2 − 10 ≈ − 0.86 and x 2 = 6 − 2 + 10 ≈ 0.19 .
Determining the Function Type Since the derivative has two distinct real roots, the derivative changes sign. This means the original function is not strictly increasing or strictly decreasing, so it is a many-to-one function.
Conclusion Therefore, the correct answer is C. It is a many-to-one function.
Examples
Many-to-one functions are common in real-world scenarios. For example, consider the function that maps a student's name to their grade in a class. Multiple students can have the same grade, making it a many-to-one function. Similarly, in signal processing, different input signals can produce the same output signal, which is also a many-to-one relationship. Understanding many-to-one functions helps in analyzing and modeling these types of relationships effectively.
