Understanding the Injectivity and Surjectivity of f(x) 2x - 1

When analyzing functions, one of the critical aspects to understand is whether a function is injective, surjective, or bijective. In this article, we will delve into the properties of the function f(x) 2x - 1. We will explore its injectivity, surjectivity, and ultimately determine if it is bijective.

Injective (One-to-One) Property

A function is considered injective or one-to-one if different inputs produce different outputs. In other words, for a function to be injective, the equation f(a) f(b) must imply that a b.

To check if f(x) 2x - 1 is injective, we start by assuming:

fa  fb

This implies:

2a - 1  2b - 1

By adding 1 to both sides and then dividing by 2, we get:

2a  2ba  b

Since a b holds true, the function f(x) 2x - 1 is injective.

Surjective (Onto) Property

A function is surjective or onto if every possible output in the codomain has a corresponding input in the domain. For our function, the codomain is the set of real numbers, ?.

To check surjectivity, we need to see if for every y ∈ ?, there exists an x ∈ ? such that:

fx  y

Starting from:

2x - 1  y

Rearranging gives:

2x  y   1x  frac{y   1}{2}

Since for every real number y, we can find a corresponding x given by:

x  frac{y   1}{2}

We conclude that the function f(x) 2x - 1 is surjective.

Conclusion

Given that the function f(x) 2x - 1 is both injective and surjective, it is bijective. A bijective function is one that is both one-to-one and onto, meaning it is both injective and surjective.

Exploring Further Properties

Let's further our analysis by examining the properties of the function f(x) x - 1.

Domain: The function is defined for all real numbers x, and is continuous.

Injective: The function is strictly increasing and monotone. Its graph is a straight line with a positive slope, which is sufficient to conclude that it is injective.

Surjective: We can prove that the function is surjective in a similar manner. For every real value y, there is at least one x such that:

y  x - 1x  y   1

This x is the counterpart of y, and it satisfies fx y.

Bijective: Since f(x) x - 1 is both injective and surjective, it is bijective.

Proving Injectivity and Surjectivity for f(x) 2x - 1 and f(x) x - 1

f(x) 2x - 1

fa  fb2x1  2y12x  2yx  y

Thus, f(x) 2x - 1 is injective.

fx  2x - 12x - 1  y2x  y   1x  frac{y   1}{2}

Thus, f(x) 2x - 1 is surjective.

f(x) x - 1

For fa fb, we have:

2x1  2y12x  2yx  y

Hence, f(x) x - 1 is injective.

fx  x - 1x - 1  yx  y   1

Thus, f(x) x - 1 is surjective.

Since both functions are injective and surjective, they are bijective.

Conclusion

In conclusion, the functions f(x) 2x - 1 and f(x) x - 1 are both bijective, meaning they are both one-to-one and onto. This property is fundamental in understanding the nature of these functions and their applications in various mathematical and practical contexts.

By analyzing the injectivity and surjectivity of these functions, we can deepen our understanding of their properties and behavior. This knowledge is crucial in fields ranging from algebra to real analysis and beyond.