To determine which statement best describes the function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex], we need to analyze the properties of this function.
1. Identify the Type of Function:
- The given function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is a quadratic function, as it is a polynomial of degree 2.
2. Check If It Is a Function:
- A function assigns exactly one output [tex]\( y \)[/tex] for each input [tex]\( x \)[/tex]. In this case, for any given [tex]\( x \)[/tex] value, there is only one corresponding [tex]\( y \)[/tex] value, as quadratic functions are continuous and defined for all real numbers [tex]\( x \)[/tex]. Hence, this is indeed a function. So, option A (It is not a function) is incorrect.
3. Check If It Is One-to-One:
- A function is one-to-one if it passes the horizontal line test, meaning that no horizontal line intersects the graph of the function at more than one point.
- Since [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is a quadratic function and its graph is a parabola that opens upwards, it does not pass the horizontal line test (a horizontal line will intersect the parabola at two points in most cases).
- Therefore, the function is not one-to-one. So, option B (It is a one-to-one function) is incorrect.
4. Check If It Is Many-to-One:
- Since a quadratic function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] can have the same [tex]\( y \)[/tex]-value for different [tex]\( x \)[/tex]-values (e.g., the same [tex]\( y \)[/tex] value for [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]), it is a many-to-one function.
- Hence, option C (It is a many-to-one function) is correct.
5. Check the Vertical Line Test:
- A function fails the vertical line test if any vertical line intersects the graph of the function at more than one point. This would mean it does not assign exactly one output for each input.
- The quadratic function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] passes the vertical line test as any vertical line will intersect the parabola at exactly one point.
- Hence, option D (It fails the vertical line test) is incorrect.
So, the best statement that describes the function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is:
C. It is a many-to-one function.
