Which statement best describes the function below?
[tex]f(x)=2 x^2-3 x+1[/tex]
A. It fails the vertical line test.
B. It is a one-to-one function.
C. It is not a function.
D. It is a many-to-one function.
Answer (2)
The function f ( x ) = 2 x 2 − 3 x + 1 is a quadratic function.
Quadratic functions pass the vertical line test, so they are functions.
Quadratic functions are symmetric, meaning multiple x values can map to the same y value.
Therefore, the function is many-to-one, and the correct answer is D .
Explanation
Analyzing the Function The given function is f ( x ) = 2 x 2 − 3 x + 1 . We need to determine which statement best describes this function from the given options.
Eliminating Option A First, let's consider option A: 'It fails the vertical line test.' The vertical line test is a visual way to determine if a curve is a graph of a function. If any vertical line intersects the curve more than once, then the curve is not a graph of a function. Since f ( x ) is a polynomial function (specifically, a quadratic function), it passes the vertical line test. Therefore, option A is incorrect.
Eliminating Option C Next, let's consider option C: 'It is not a function.' By definition, a function must pass the vertical line test. Since f ( x ) is a quadratic function and passes the vertical line test, it is indeed a function. Thus, option C is incorrect.
Determining One-to-One or Many-to-One Now, let's analyze options B and D. Option B states: 'It is a one-to-one function.' A one-to-one function means that for every y value, there is only one x value. Option D states: 'It is a many-to-one function.' A many-to-one function means that for a given y value, there are multiple x values that map to the same y value. Quadratic functions are parabolas, which are symmetric about their vertex. This symmetry implies that for most y values (except the y value at the vertex), there are two different x values that produce the same y value. For example, let's find x 1 and x 2 such that f ( x 1 ) = f ( x 2 ) .
Consider x 1 = 0 :
f ( 0 ) = 2 ( 0 ) 2 − 3 ( 0 ) + 1 = 1
Now, let's find another x value, x 2 , such that f ( x 2 ) = 1 :
2 x 2 2 − 3 x 2 + 1 = 1 2 x 2 2 − 3 x 2 = 0 x 2 ( 2 x 2 − 3 ) = 0 So, x 2 = 0 or 2 x 2 − 3 = 0 , which means x 2 = 2 3 = 1.5 Since f ( 0 ) = f ( 1.5 ) = 1 , the function is many-to-one.
Conclusion Since the function is many-to-one, option D is the correct answer.
Examples
Many-to-one functions are common in real life. For example, consider a grading system where multiple students can score different marks (x values) but end up with the same grade (y value). Another example is the height of people; many people can have different names and identities (x values) but share the same height (y value). Quadratic functions, like the one in this problem, are a fundamental example of many-to-one relationships due to their symmetrical nature.
The function f ( x ) = 2 x 2 − 3 x + 1 is a quadratic function that passes the vertical line test, confirming it is indeed a function. It is a many-to-one function because multiple x values can yield the same y value, such as f ( 0 ) = f ( 1.5 ) = 1 . Therefore, the correct choice is D .
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