To determine which statement best describes the function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex], we will analyze its properties step-by-step.
1. Identify the Type of Function:
The given function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is a quadratic function. Quadratic functions are polynomial functions of degree 2.
2. Vertical Line Test:
The vertical line test is used to determine if a graph represents a function. If any vertical line intersects the graph of the function more than once, the graph does not represent a function.
- Quadratic functions always pass the vertical line test because any vertical line will intersect the graph of [tex]\( f(x) \)[/tex] at most once. Therefore, this function is indeed a function.
3. One-to-One vs. Many-to-One:
- A one-to-one function means that each y-value is mapped by exactly one x-value.
- A many-to-one function means that some y-values are mapped by more than one x-value.
To determine if [tex]\( f(x) \)[/tex] is one-to-one or many-to-one, let's consider the nature of a quadratic function. Quadratic functions have a parabolic shape and are symmetric about their vertex. Because of this symmetry, the same y-value can often be achieved by two different x-values.
For instance, consider the quadratic function [tex]\( f(x) \)[/tex]. If you compute [tex]\( f(x) \)[/tex] at [tex]\( x = x_1 \)[/tex] and [tex]\( x = x_2 \)[/tex] such that [tex]\( x_1 \neq x_2 \)[/tex], it is possible that [tex]\( f(x_1) = f(x_2) \)[/tex]. This property indicates that quadratic functions are generally many-to-one.
Given these observations:
- It is a function (passes the vertical line test).
- It is many-to-one rather than one-to-one.
Therefore, the best description is:
A. It is a many-to-one function.
