Function Evaluation From Ordered Pairs Identifying The Correct Statement

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In the fascinating world of mathematics, functions play a crucial role in describing relationships between variables. A function, at its core, is a rule that assigns to each input value exactly one output value. These relationships can be represented in various ways, including equations, graphs, and sets of ordered pairs. In this article, we will delve into the concept of functions defined by ordered pairs and explore how to determine the truth of statements related to them. Understanding functions is a fundamental concept in mathematics, serving as the backbone for more advanced topics like calculus and analysis. When a function is defined by a set of ordered pairs, it explicitly lists the inputs and their corresponding outputs. Each ordered pair has the format (input, output), where the input is often denoted as 'x' and the output as 'f(x)'. The function essentially acts as a mapping from the set of inputs (domain) to the set of outputs (range). When analyzing a function represented by ordered pairs, our goal is often to evaluate the function at specific input values or to determine the function's properties based on the given data points. This may involve checking if a particular input-output pair belongs to the function, identifying the function's domain and range, or even attempting to find an equation that represents the function's behavior. The ability to interpret and work with functions presented in this form is essential for solving a wide range of mathematical problems and real-world applications. Whether it's understanding the relationship between time and distance in physics, analyzing financial trends, or modeling population growth, functions defined by ordered pairs provide a direct and intuitive way to represent dependencies between quantities. As we explore the given function in this article, we'll see how these concepts come into play and how we can use ordered pairs to unlock the secrets of a function's behavior.

The Function f(x) and its Ordered Pairs

In this particular problem, we are presented with a function, denoted as f(x), which is defined by a set of ordered pairs. Ordered pairs are a fundamental way to represent the relationship between an input and an output in a function. The set of ordered pairs for our function is given as: {(1,0), (-10,2), (0,6), (3,17), (-2,-1)}. Each ordered pair in this set represents a specific input-output relationship. For example, the ordered pair (1,0) tells us that when the input is 1, the output of the function f(x) is 0. Similarly, the ordered pair (-10,2) indicates that when the input is -10, the output is 2. To fully understand the function, we must carefully interpret each ordered pair. The first value in each pair is the input, often referred to as the 'x' value or the independent variable. The second value is the output, often referred to as the 'f(x)' value or the dependent variable, as its value depends on the input. Understanding these input-output relationships is crucial for analyzing the function's behavior. By examining the set of ordered pairs, we can start to get a sense of how the function transforms inputs into outputs. We can see specific instances where the function takes on certain values, and this can help us to identify patterns or potentially infer an equation that describes the function. For instance, the pair (0,6) tells us that the function has a value of 6 when the input is 0. This is the y-intercept of the function if we were to graph it. The ordered pairs (-2,-1) and (3,17) give us two more points on the function, and we can use these points, along with the others, to make observations about the function's behavior. In this exercise, we're going to be asked to evaluate the function at particular points, using the information provided by the ordered pairs. This process involves matching the input value to the corresponding output value in the given set. It's a direct way to understand what the function 'does' at specific points and to verify statements about the function's values. Mastering the interpretation of ordered pairs is key to understanding functions and their applications in various mathematical and real-world scenarios.

Evaluating the Given Statements

Now that we have the function f(x) defined by the set of ordered pairs {(1,0), (-10,2), (0,6), (3,17), (-2,-1)}, our next step is to evaluate the given statements to determine which one is true. Evaluating function statements based on ordered pairs involves checking if the input-output relationship claimed in the statement matches the information provided in the set of ordered pairs. Each statement presents a potential input-output relationship for the function f(x). To determine the truth of a statement, we will look for the ordered pair that corresponds to the input value in the statement. If we find an ordered pair with the matching input value, we then compare the stated output value with the output value in the ordered pair. If the output values match, the statement is true; otherwise, it is false. Let's consider each statement one by one:

  • Statement A: f(-10) = 1 To evaluate this statement, we look for an ordered pair in the set where the input value is -10. We find the ordered pair (-10, 2). This tells us that f(-10) is equal to 2, not 1. Therefore, statement A is false.
  • Statement B: f(2) = -10 For this statement, we need to find an ordered pair with an input value of 2. However, a careful examination of the set of ordered pairs reveals that there is no pair with an input of 2. Therefore, we cannot determine the value of f(2) from the given information. Since we do not have an ordered pair to support the statement that f(2) = -10, we must conclude that statement B is false. Even if the function might have a value at x=2, we cannot confirm it based on the provided data.
  • Statement C: f(0) = 6 To verify this statement, we search for an ordered pair with an input of 0. We find the ordered pair (0, 6). This pair indicates that when the input is 0, the output of the function is 6. Thus, the statement f(0) = 6 is true.
  • Statement D: f(1) = -10 To assess this statement, we look for an ordered pair with an input value of 1. We find the ordered pair (1, 0). This pair tells us that f(1) is equal to 0, not -10. Therefore, statement D is false.

Through this process of comparing each statement to the provided ordered pairs, we can systematically determine which statement accurately reflects the function's behavior.

Determining the Correct Equation

After evaluating each statement, we can now confidently determine which equation is true. As we discussed in the previous section, we examined each statement by comparing it to the given set of ordered pairs: {(1,0), (-10,2), (0,6), (3,17), (-2,-1)}. Our analysis revealed the following:

  • Statement A: f(-10) = 1 is false, because the ordered pair (-10, 2) indicates that f(-10) = 2.
  • Statement B: f(2) = -10 is false, because there is no ordered pair with an input of 2 in the given set, so we cannot determine f(2) from this data.
  • Statement C: f(0) = 6 is true, because the ordered pair (0, 6) explicitly shows that when the input is 0, the output is 6.
  • Statement D: f(1) = -10 is false, because the ordered pair (1, 0) indicates that f(1) = 0.

Based on this evaluation, we can definitively conclude that only statement C, f(0) = 6, accurately reflects the information provided by the ordered pairs. Determining the correct equation from a set of options requires a careful comparison of each statement against the given data. In this case, the set of ordered pairs serves as the ground truth for the function's behavior. By checking each proposed relationship between input and output against the ordered pairs, we can eliminate incorrect options and identify the one that aligns perfectly with the function's definition. This process highlights the importance of understanding how functions are represented by ordered pairs and how to interpret these pairs to evaluate function values. The ability to correctly evaluate function statements is a crucial skill in mathematics, as it forms the basis for understanding more complex functional relationships and problem-solving techniques. Whether we are working with linear functions, quadratic functions, or any other type of function, the fundamental principle of matching inputs to their corresponding outputs remains the same. In this specific example, the correct equation is readily identified through direct comparison with the ordered pairs, but in more complex scenarios, this evaluation process may involve additional steps, such as algebraic manipulation or graphical analysis.

Conclusion

In conclusion, by carefully examining the function f(x) defined by the set of ordered pairs {(1,0), (-10,2), (0,6), (3,17), (-2,-1)} and systematically evaluating the given statements, we have determined that the equation f(0) = 6 is the only true statement. This exercise underscores the importance of understanding how functions are represented by ordered pairs and how to interpret these pairs to evaluate function values. Understanding functions is a cornerstone of mathematics, and the ability to work with functions presented in various forms is crucial for solving a wide range of problems. Functions, as we've seen, are a way to describe the relationship between inputs and outputs, and ordered pairs provide a direct and explicit representation of this relationship. When a function is defined by a set of ordered pairs, each pair tells us exactly what the output is for a specific input. This directness allows us to verify statements about the function by simply checking if the stated input-output relationship is present in the set of pairs. In the context of this problem, we used this method to evaluate statements involving function values at specific points. We looked for ordered pairs with the given input values and compared the corresponding output values to those stated in the equations. This process allowed us to quickly identify the correct equation and rule out the incorrect ones. The systematic approach we used – evaluating each statement individually against the given data – is a valuable problem-solving technique that can be applied in many different mathematical contexts. Whether we're dealing with simple functions defined by ordered pairs or more complex functions defined by equations or graphs, the fundamental principle of checking input-output relationships remains the same. By mastering this principle, we can confidently navigate the world of functions and unlock their power to model and solve real-world problems.