Find The Value Of C For A One-to-One Function

To determine the value of c that makes the given function one-to-one, it’s crucial to first understand the fundamental concept of what a one-to-one function, also known as an injective function, truly represents. In mathematical terms, a function is considered one-to-one if each element of the range corresponds to exactly one element of the domain. Put simply, this means that no two different inputs (x-values) produce the same output (y-value). This property is critical in various mathematical contexts, including inverse functions and cryptography.

The Key Characteristic: Distinct Outputs for Distinct Inputs The defining characteristic of a one-to-one function is that if f(x₁) = f(x₂), then x₁ must equal x₂. In simpler terms, if two different inputs produce the same output, the function is not one-to-one. Conversely, if every distinct input results in a distinct output, the function is one-to-one. This distinctiveness is what ensures that the function can be uniquely reversed, which is a key property for inverse functions. A visual way to determine if a function is one-to-one is by using the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, the function is not one-to-one. This is because the points of intersection represent inputs with the same output.

Practical Implications and Examples Consider the function f(x) = x². This function is not one-to-one because, for example, f(2) = 4 and f(-2) = 4. Two different inputs, 2 and -2, produce the same output, 4, violating the one-to-one property. On the other hand, the function f(x) = x³ is one-to-one because every real number has a unique cube, and every cube has a unique real root. This distinction highlights the importance of understanding the behavior of different types of functions when determining whether they are one-to-one.

Why One-to-One Functions Matter The concept of one-to-one functions is not just a theoretical exercise; it has practical implications in various fields. In cryptography, one-to-one functions are used to ensure that each encryption has a unique decryption, which is crucial for secure communication. In data science, one-to-one mappings can be used to ensure that data transformations preserve the uniqueness of the original data. Understanding one-to-one functions is also essential in calculus, particularly when dealing with inverse functions and transformations.

Now, let's delve into the specifics of the problem at hand. We are given a function defined as a set of ordered pairs: {(1, 2), (2, 3), (3, 5), (4, 7), (5, 11), (6, c)}. To determine the value of c that makes this function one-to-one, we need to ensure that no two different x-values (inputs) map to the same y-value (output). This is the fundamental principle of one-to-one functions, and it is the key to solving this problem.

Examining the Existing Ordered Pairs First, let's analyze the existing ordered pairs to identify the current range of the function. We have the following mappings:

• 1 maps to 2
• 2 maps to 3
• 3 maps to 5
• 4 maps to 7
• 5 maps to 11

This gives us a current range of {2, 3, 5, 7, 11}. For the function to be one-to-one, the value of c must not be any of these existing y-values. If c were equal to any of these numbers, then the function would have two different x-values mapping to the same y-value, thus violating the one-to-one property. For instance, if c were 2, then we would have both (1, 2) and (6, 2), which means the function is no longer one-to-one.

Identifying the Constraint on c The crucial constraint on c is that it cannot be any of the existing y-values in the range. This means c cannot be 2, 3, 5, 7, or 11. The value of c must be a number that is not already associated with any of the x-values in the domain. This is the essence of maintaining the one-to-one property. Each x-value must have a unique y-value, and each y-value must be associated with only one x-value.

The Importance of Uniqueness in Mappings The uniqueness of mappings is what distinguishes a one-to-one function from other types of functions. In a one-to-one function, each input has a unique output, and each output has a unique input. This property is essential for the existence of an inverse function. If a function is not one-to-one, its inverse will not be a function because there will be some outputs that map to multiple inputs, violating the definition of a function. Therefore, maintaining the uniqueness of mappings is critical for various mathematical operations and applications.

Now that we understand the constraints on c, let's look at the options provided to determine which value makes the function one-to-one. The options are A. 2, B. 5, C. 11, and D. 13. We've already established that c cannot be 2, 5, or 11 because these values are already in the range of the function.

Evaluating the Options

• A. 2: If c = 2, the function would have the ordered pairs (1, 2) and (6, 2). This means two different inputs (1 and 6) map to the same output (2), violating the one-to-one property.
• B. 5: If c = 5, the function would have the ordered pairs (3, 5) and (6, 5). Again, two different inputs (3 and 6) map to the same output (5), which means the function is not one-to-one.
• C. 11: If c = 11, the function would have the ordered pairs (5, 11) and (6, 11). Two different inputs (5 and 6) map to the same output (11), violating the one-to-one property.
• D. 13: If c = 13, the function would have the ordered pair (6, 13). Since 13 is not already in the range of the function, this value does not violate the one-to-one property. Each input has a unique output, and each output has a unique input.

The Solution: c = 13 Therefore, the only value of c that makes the function one-to-one is 13. When c is 13, the function is defined as {(1, 2), (2, 3), (3, 5), (4, 7), (5, 11), (6, 13)}. In this function, each x-value maps to a unique y-value, and each y-value is associated with only one x-value. This satisfies the fundamental requirement for a function to be one-to-one.

Ensuring the One-to-One Property To reiterate, the key to ensuring a function is one-to-one is to verify that no two distinct inputs produce the same output. By carefully examining the ordered pairs and ensuring that the value of c does not duplicate any existing y-values in the range, we can maintain the one-to-one property. This principle is not only important for solving this specific problem but also for understanding more complex mathematical concepts related to functions and their inverses.

The correct answer is D. 13. This is the only value of c that ensures the function remains one-to-one because it does not duplicate any existing y-values in the range. This maintains the uniqueness of the mappings, which is the defining characteristic of a one-to-one function.