One-to-One Function: Find The Value Of 'c'

by Alex Johnson 43 views

Let's dive into the world of one-to-one functions and figure out how to determine the value of 'c' that makes the given function fit the bill. This is a classic problem in mathematics that tests our understanding of function properties. We'll break down the concept, analyze the function, and arrive at the correct answer. Let's make this journey through functions both informative and fun!

Understanding One-to-One Functions

At its heart, a one-to-one function, also known as an injective function, has a unique output for every unique input. Think of it like a perfectly matched dating app where each person (input) is paired with only one other person (output), and vice versa. Mathematically, this means if f(x₁) = f(x₂), then x₁ must equal x₂. In simpler terms, no two different inputs can produce the same output. This is the key concept we need to keep in mind as we tackle our problem. Understanding this concept is crucial, not just for solving this specific problem but for grasping many other concepts in mathematics and related fields. This principle appears in various forms, from basic algebra to advanced calculus, and even in fields like cryptography and computer science.

To visualize this, imagine a graph. A function is one-to-one if it passes the horizontal line test. This means that if you draw any horizontal line across the graph, it will intersect the function's line at most once. If the line intersects more than once, it indicates that different inputs are mapping to the same output, thus violating the one-to-one property. Understanding this visual representation can greatly aid in quickly identifying whether a function is one-to-one, especially when dealing with graphical representations of functions. The horizontal line test provides a simple, yet powerful, tool for analyzing functions at a glance. This is particularly useful in situations where an algebraic approach might be cumbersome or time-consuming.

Now, why is this concept so important? One-to-one functions have inverses. The inverse function essentially reverses the mapping, taking the output back to its unique input. This is a fundamental concept in many areas of mathematics, including solving equations, understanding transformations, and even in more advanced topics like linear algebra and functional analysis. The existence of an inverse function is predicated on the function being one-to-one. If a function is not one-to-one, it cannot have a well-defined inverse because there would be ambiguity in mapping the output back to the input. This is a critical point to understand, as it highlights the significance of one-to-one functions in the broader mathematical landscape. So, with the definition firmly in our minds, let's apply it to the problem at hand and see how we can determine the value of 'c' to ensure our function is indeed one-to-one.

Analyzing the Given Function

We're presented with a function defined as a set of ordered pairs: {(1,2), (2,3), (3,5), (4,7), (5,11), (6, c)}. Each pair represents an input (x) and its corresponding output (f(x)). To ensure this function is one-to-one, the output values must all be unique. We need to find the value of 'c' that doesn't repeat any of the existing output values. Let’s take a close look at the outputs we already have: 2, 3, 5, 7, and 11. Our task is to determine which of the given options for 'c' will maintain the uniqueness of the outputs. This step involves careful observation and comparison, ensuring that the chosen value for 'c' does not create a duplicate output, which would violate the one-to-one function requirement.

This part of the problem emphasizes the importance of paying attention to detail. A simple oversight can lead to an incorrect conclusion. The beauty of this type of problem lies in its directness; the solution is often quite straightforward once the underlying concept is clearly understood. It's a matter of applying the definition of a one-to-one function in a practical context. By carefully examining the existing outputs and comparing them against the potential values of 'c', we can systematically narrow down the possibilities and arrive at the correct answer. This process highlights the analytical skills needed in mathematics, where careful observation and logical deduction are paramount. Furthermore, it underscores the connection between theory and practice, demonstrating how abstract concepts like one-to-one functions have concrete implications in problem-solving scenarios. So, with our eyes peeled and our minds focused, let's proceed to evaluate the given options and pinpoint the value of 'c' that keeps our function in the realm of one-to-one functions.

Looking at the output values (2, 3, 5, 7, 11), we need to choose a value for 'c' that is different from all of these. This ensures that no two different inputs map to the same output. The essence of a one-to-one function lies in this unique mapping – each input has its own exclusive output, and vice versa. Think of it as a highly exclusive club where each member has their own unique code to enter, and no two members share the same code. In our function, each input (x-value) has a corresponding output (y-value), and for the function to be one-to-one, these outputs must be distinct. This is not just a mathematical requirement; it has real-world analogies. For example, in database management, primary keys need to be unique to ensure data integrity. Similarly, in cryptography, unique keys are essential for secure communication. Understanding the principle of unique mapping is thus applicable beyond the realm of pure mathematics.

Evaluating the Options

Now let's consider the given options for 'c':

  • A. 2: This value is already present in the output set, so it would make the function not one-to-one.
  • B. 5: Similar to 2, this value also exists in the output set, disqualifying it.
  • C. 11: This value is also a duplicate, making it an invalid choice.
  • D. 13: This value is not in the existing output set. This means that if we set c = 13, there will be no two inputs mapping to the same output.

Therefore, the correct answer is D. 13. By carefully comparing each option with the existing outputs, we have identified the value of 'c' that preserves the one-to-one nature of the function. This process not only solves the problem at hand but also reinforces our understanding of the fundamental concept of one-to-one functions. It's a testament to the power of systematic analysis and the importance of attention to detail in mathematical problem-solving. Moreover, it highlights how mathematical principles can be applied in a step-by-step manner to arrive at a logical conclusion. So, with the value of 'c' determined, we've successfully navigated the intricacies of one-to-one functions and demonstrated our ability to apply theoretical knowledge to practical problem-solving.

Conclusion

In conclusion, the value of 'c' that makes the function one-to-one is 13. This ensures that each input has a unique output, satisfying the condition for a one-to-one function. We've not only solved the problem but also reinforced our understanding of the concept of one-to-one functions, which is a valuable takeaway for future mathematical endeavors. Remember, the key to mastering mathematics lies not just in memorizing formulas but in understanding the underlying principles and applying them effectively. This problem serves as a perfect example of how a solid grasp of fundamental concepts can lead to elegant solutions. So, keep practicing, keep exploring, and keep the fascinating world of mathematics alive!

For further exploration of one-to-one functions, you can visit resources like Khan Academy's article on injective functions.