Introduction

In the realm of mathematics, a function f(x) can be classified into three categories based on its properties: injective (one-to-one), surjective (onto), and bijective (one-to-one and onto). To elucidate these concepts, let's explore the function f(x) 2x and determine its properties.

Key Definitions

Injective (One-to-One): A function is injective if different inputs map to different outputs. In other words, for any two distinct elements a and b in the domain, if fa fb, then a b. Surjective (Onto): A function is surjective if every possible output in the codomain is achieved by at least one input in the domain. For any value y in the codomain, there exists at least one x in the domain such that fx y. Bijective: A function is bijective if it is both injective and surjective, meaning it is one-to-one and onto.

Case Study: f(x) 2x

To determine whether the function f(x) 2x is injective, surjective, or bijective, let's analyze its properties step by step.

Injectivity: One-to-One

A function f(x) is injective if no two distinct inputs yield the same output. For the function f(x) 2x, assume fa fb. Given fa 2a and fb 2b, if fa fb, then 2a 2b. Dividing both sides by 2, we get a b. Therefore, the function is injective.

Surjectivity: Onto

A function f(x) is surjective if for every value in the codomain, there exists at least one input in the domain that maps to it. To test this, let's assume the codomain is the set of real numbers #x1D442;.

Given the equation f(x) 2x, we need to find an x for any given real number y such that fx y. Solving for x, we get x y/2. Since y is a real number, x will also be a real number. Therefore, for any y in #x1D442;, there exists an x in #x1D442; such that fx y. Hence, the function is surjective.

Bijectivity: One-to-One and Onto

A function is bijective if it is both injective and surjective. Since we have established that f(x) 2x is both injective and surjective, it is bijective.

Conclusion

Thus, the function f(x) 2x is both injective and surjective, making it bijective. Understanding the definitions and properties of injective, surjective, and bijective functions is crucial for solving problems related to function mappings.

Additional Notes

It's important to note that the properties of functions can change based on the specified domain and codomain. For instance, if the function f(x) 2x maps integers to integers, it would not be surjective because there would be no integer x that maps to an odd integer. However, with the domain and codomain being the set of all real numbers, the function is both injective and surjective, making it bijective.

Mastery of these definitions and the ability to apply them allows for quick and accurate assessments of function properties. As with the problem presented, if one is familiar with the definitions, the solution can often be determined within seconds, demonstrating the importance of memorization and comprehension in mathematical problem-solving.