Answer :
To determine which statement best describes the function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex], let's analyze its characteristics step-by-step.
1. Identify the type of function:
- The function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is a quadratic function because it is a polynomial with a degree of 2. The general form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex] where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
2. Determine if it is a one-to-one function:
- A function is one-to-one if and only if every value of [tex]\( y \)[/tex] corresponds to exactly one value of [tex]\( x \)[/tex]. For quadratic functions, they are generally not one-to-one because they are parabolic in shape (they have a U-shaped curve). This means that for some [tex]\( y \)[/tex]-values, there can be two different [tex]\( x \)[/tex]-values that produce the same [tex]\( y \)[/tex]-value. Therefore, [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is not a one-to-one function.
3. Determine if it is a function:
- A relation is a function if each input [tex]\( x \)[/tex] has exactly one output [tex]\( y \)[/tex]. Quadratic functions like [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] pass the vertical line test, meaning that any vertical line drawn on the graph will intersect the curve at most once. Thus, [tex]\( f(x) \)[/tex] is indeed a function.
4. Determine if it is a many-to-one function:
- A many-to-one function is a function where multiple [tex]\( x \)[/tex]-values can map to the same [tex]\( y \)[/tex]-value. As discussed earlier, quadratic functions generally have this property. In [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex], different [tex]\( x \)[/tex]-values can produce the same [tex]\( y \)[/tex]-value, confirming that it is a many-to-one function.
5. Determine if it fails the vertical line test:
- The vertical line test is used to determine if a curve is a function. If any vertical line intersects the graph of the relation more than once, then the relation is not a function. Since [tex]\( f(x) \)[/tex] passes the vertical line test, it does not fail this test.
Based on this analysis, the correct statement that best describes the function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is:
C. It is a many-to-one function.
1. Identify the type of function:
- The function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is a quadratic function because it is a polynomial with a degree of 2. The general form of a quadratic function is [tex]\( ax^2 + bx + c \)[/tex] where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants.
2. Determine if it is a one-to-one function:
- A function is one-to-one if and only if every value of [tex]\( y \)[/tex] corresponds to exactly one value of [tex]\( x \)[/tex]. For quadratic functions, they are generally not one-to-one because they are parabolic in shape (they have a U-shaped curve). This means that for some [tex]\( y \)[/tex]-values, there can be two different [tex]\( x \)[/tex]-values that produce the same [tex]\( y \)[/tex]-value. Therefore, [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is not a one-to-one function.
3. Determine if it is a function:
- A relation is a function if each input [tex]\( x \)[/tex] has exactly one output [tex]\( y \)[/tex]. Quadratic functions like [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] pass the vertical line test, meaning that any vertical line drawn on the graph will intersect the curve at most once. Thus, [tex]\( f(x) \)[/tex] is indeed a function.
4. Determine if it is a many-to-one function:
- A many-to-one function is a function where multiple [tex]\( x \)[/tex]-values can map to the same [tex]\( y \)[/tex]-value. As discussed earlier, quadratic functions generally have this property. In [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex], different [tex]\( x \)[/tex]-values can produce the same [tex]\( y \)[/tex]-value, confirming that it is a many-to-one function.
5. Determine if it fails the vertical line test:
- The vertical line test is used to determine if a curve is a function. If any vertical line intersects the graph of the relation more than once, then the relation is not a function. Since [tex]\( f(x) \)[/tex] passes the vertical line test, it does not fail this test.
Based on this analysis, the correct statement that best describes the function [tex]\( f(x) = 2x^2 - 3x + 1 \)[/tex] is:
C. It is a many-to-one function.
