Determining Bijection of Quadratic Functions: A Case Study with fx x2 - 3x - 2

The concept of a bijection is fundamental in mathematics, particularly concerning the properties of functions. A function is considered a bijection if it is both injective (one-to-one) and surjective (onto). In this article, we will delve into the analysis of the function fx x2 - 3x - 2 to determine whether it is a bijection.

Introduction to the Function

Assume we have the function fx x2 - 3x - 2, which is a quadratic function. Our goal is to examine whether this function is a bijection by verifying its injective and surjective properties.

Injectivity (One-to-One)

A function is injective if any two inputs produce distinct outputs. Mathematically, fa fb implies a b.

Analysis

To test for injectivity, we can compute the derivative fx' to understand the behavior of the function. The derivative of fx is

fx' 2x - 3

Setting the derivative equal to zero to find the critical points, we have

2x - 3 0 which gives x 3/2

The function fx x2 - 3x - 2 is a quadratic function with a positive leading coefficient, indicating that it opens upwards. At the vertex, the function reaches its minimum value. Since a quadratic function that opens upwards is not one-to-one, it takes the same value at two different points on either side of the vertex. Therefore, fx is not injective.

Surjectivity (Onto)

A function is surjective if every element in the codomain is mapped to by at least one element in the domain. We need to check if the range of the function covers all possible real numbers.

Analysis

The minimum value of the function fx occurs at the vertex. The vertex of the quadratic function takes place where the derivative is zero, which we have already determined to be x 3/2. Plugging this value into the function, we get

f(3/2) (3/2 - 1)(3/2 - 2) (1/2)(-1/2) -1/4

Since the function opens upwards and the minimum value is -1/4, the range of the function is [-1/4, ∞).

If the codomain is the set of all real numbers R, then not all real numbers are in the range of fx. Specifically, values less than -1/4 are not covered by the function. Therefore, fx is not surjective.

Conclusion

Since the function fx x2 - 3x - 2 fails to be both injective and surjective, it is not a bijection.

Alternative Analysis

We can also verify our conclusion using an alternative approach. A function is not one-to-one (injective) if there exist at least two distinct inputs that produce the same output. Specifically, we can show that if fx1 fx2 for (f(x_1) 0) and (f(x_2) 0), then the function is not injective.

Alternative Proofs of Non-One-to-One Property

We see that ((x - 1)(x - 2) 0) implies (x 1) or (x 2). Hence, (f(1) f(2) 0), which violates the one-to-one property.

Non-Surjective Property

To prove that the function is not surjective, we observe that there exist some real numbers that are not in the range of the function. Specifically, consider (y -2). If (f(x) -2) for some (x), then

(x^2 - 3x - 2 -2)

(x^2 - 3x 0)

(x(x - 3) 0)

(x 0 text{ or } x 3)

These solutions are real, but the function can also have the form (x frac{-3 pm sqrt{9 16}}{2} frac{-3 pm sqrt{25}}{2} frac{-3 pm 5}{2}), which gives (x 1) or (x -2). However, the function does not cover all values less than (-frac{1}{4}).

Final Conclusion

Since the function (fx x^2 - 3x - 2) is neither injective nor surjective, it is not a bijection.