One-to-One, Onto, And Pre-image Analysis Of F(x) = X + 3
In the realm of mathematics, functions serve as fundamental building blocks for modeling relationships between sets. Understanding the properties of a function, such as whether it is one-to-one (injective), onto (surjective), and its pre-image, provides valuable insights into its behavior and applications. This article delves into the analysis of the function f(x) = x + 3, where the domain and codomain are the set of natural numbers (N). We will investigate whether this function is one-to-one and onto, and we will also determine the pre-image of the number 100.
Determining if f(x) = x + 3 is One-to-One (Injective)
One-to-one functions, also known as injective functions, are functions where each element in the codomain is mapped to by at most one element in the domain. In simpler terms, this means that if two different inputs produce the same output, then the function is not one-to-one. To formally prove that a function is one-to-one, we need to show that if f(x₁) = f(x₂) , then x₁ = x₂. Let's apply this concept to our function f(x) = x + 3.
Assume that f(x₁) = f(x₂) for some x₁ and x₂ in the set of natural numbers N. Then, according to the definition of our function, we have:
x₁ + 3 = x₂ + 3
To determine if the function is one-to-one, we need to determine if this equation implies that x₁ = x₂. Subtracting 3 from both sides of the equation, we get:
x₁ = x₂
This result shows that if f(x₁) = f(x₂) , then it must be the case that x₁ = x₂. Therefore, the function f(x) = x + 3 satisfies the condition for being one-to-one. This is because each natural number x in the domain is mapped to a unique value x + 3 in the codomain. No two different natural numbers will produce the same output when the function is applied to them. This one-to-one property is crucial in many mathematical contexts, as it guarantees that each element in the range has a unique antecedent in the domain.
Determining if f(x) = x + 3 is Onto (Surjective)
Onto functions, also known as surjective functions, are functions where every element in the codomain is mapped to by at least one element in the domain. In other words, for every element y in the codomain, there must exist an element x in the domain such that f(x) = y. To determine if our function f(x) = x + 3 is onto, we need to investigate whether every natural number in the codomain has a corresponding natural number in the domain that maps to it.
Let y be an arbitrary element in the codomain, which is the set of natural numbers N. We want to find an element x in the domain (also N) such that f(x) = y. Using the definition of our function, this means we need to solve the equation:
x + 3 = y
for x. Subtracting 3 from both sides, we get:
x = y - 3
Now, here's where the challenge arises. While the equation x = y - 3 gives us a potential value for x, we need to ensure that this value is also a natural number. Recall that the set of natural numbers N typically includes the positive integers (1, 2, 3, ...) and may or may not include 0, depending on the definition used. Let's consider the case where y = 1. Then,
x = 1 - 3 = -2
The result, x = -2, is not a natural number. This demonstrates that there exists an element in the codomain (namely, 1) for which there is no corresponding natural number in the domain that maps to it. Therefore, the function f(x) = x + 3 is not onto when the domain and codomain are the set of natural numbers. The lack of the onto property indicates that the function does not cover the entire codomain. There are elements in the codomain that are not the image of any element in the domain.
Finding the Pre-image of 100 for f(x) = x + 3
The pre-image of an element y in the codomain is the set of all elements x in the domain that map to y under the function f. In other words, it is the set of all x such that f(x) = y. To find the pre-image of 100 for our function f(x) = x + 3, we need to find all natural numbers x that satisfy the equation:
f(x) = 100
Substituting the definition of our function, we get:
x + 3 = 100
To solve for x, we subtract 3 from both sides of the equation:
x = 100 - 3
x = 97
Since 97 is a natural number, it is indeed a valid element in the domain. Therefore, the pre-image of 100 for the function f(x) = x + 3 is the set containing the single element {97}. This means that the only natural number that maps to 100 under this function is 97. The pre-image concept is useful for understanding the inverse relationship of a function and for solving equations involving the function.
Conclusion
In summary, our analysis of the function f(x) = x + 3 when the domain and codomain are the set of natural numbers (N) has revealed the following: The function is one-to-one (injective), meaning that each element in the codomain is mapped to by at most one element in the domain. The function is not onto (surjective), meaning that there are elements in the codomain that are not the image of any element in the domain. The pre-image of 100 for this function is the set {97}, indicating that 97 is the only natural number that maps to 100 under this function. Understanding these properties of functions is essential for various mathematical applications, including cryptography, computer science, and mathematical modeling. By determining whether a function is one-to-one and onto, and by finding pre-images, we gain a deeper understanding of the function's behavior and its relationship between the domain and codomain.
