Evaluating Functions For A Given Domain A Step By Step Guide

In mathematics, understanding functions and their domains is crucial. A function is essentially a rule that assigns each input value (from the domain) to a unique output value. The domain of a function is the set of all possible input values for which the function is defined. In simpler terms, it's the list of x-values you're allowed to plug into the function. Once we have the function and its domain defined, the next step is often to evaluate the function for specific values within that domain. This involves substituting the input values into the function's equation and calculating the corresponding output values, also known as the y-values. Let's explore this process with a concrete example. Consider a function defined by the equation y = -2/3 x + 7, and suppose we are given the domain as {-12, -6, 3, 15}. This means we need to find the y-values that correspond to each of these x-values. The table completion becomes a straightforward task of plugging in each x-value from the domain into the equation and solving for y. This practical approach not only reinforces the understanding of functions and domains but also solidifies algebraic skills in evaluating expressions. The concept of a function's domain is fundamental in mathematics because it defines the boundaries within which the function operates meaningfully. Attempting to evaluate a function outside its domain can lead to undefined results or mathematical inconsistencies. Therefore, recognizing and understanding the domain is a critical first step in any function-related problem. Once the domain is established, the process of evaluating the function for specific values within that domain becomes a matter of careful substitution and arithmetic calculation. Each x-value from the domain is plugged into the function's equation, and the corresponding y-value is determined. This pairing of x- and y-values then forms a coordinate point that lies on the graph of the function. By evaluating the function for multiple values within the domain, we can begin to map out the behavior and shape of the function. For linear functions, like the one we're using as an example, plotting these points allows us to visualize the straight line that represents the function's graph. For more complex functions, this process of evaluation and plotting helps to understand the function's curves, turning points, and overall trend. In essence, evaluating functions for values within their domains is a cornerstone of mathematical analysis and provides a tangible way to connect algebraic expressions with their graphical representations.

Step-by-Step Evaluation of the Function

H2: Evaluating y when x = -12

Let’s start with the first value in the domain, x = -12. To find the corresponding y-value, we substitute -12 for x in the equation y = -2/3 x + 7. This gives us y = -2/3 * (-12) + 7. Now, we perform the multiplication: -2/3 multiplied by -12 equals 8. So, the equation becomes y = 8 + 7. Adding these numbers, we find that y = 15. Thus, when x is -12, y is 15. This calculation provides the first coordinate pair (-12, 15), which represents a point on the graph of the function. This process of substitution and simplification is a fundamental skill in algebra and is essential for working with functions. By carefully replacing the variable x with its given value, we can solve for the corresponding y-value, which represents the output of the function for that particular input. The order of operations (PEMDAS/BODMAS) must be followed meticulously to ensure accurate results. In this case, we performed the multiplication before the addition, adhering to the standard algebraic conventions. The resulting y-value, 15, completes the first entry in our table and provides a crucial piece of information about the behavior of the function. Each point calculated in this manner contributes to our understanding of the function's graph and its overall characteristics. The ability to accurately evaluate functions for specific input values is not only a cornerstone of mathematical problem-solving but also a valuable skill in various scientific and engineering applications. Whether modeling physical phenomena, analyzing data, or designing algorithms, the ability to translate inputs into outputs using mathematical functions is a fundamental necessity. As we continue to evaluate the function for other values in the domain, we'll further solidify our understanding of its behavior and the relationship between x and y.

H2: Evaluating y when x = -6

Next, we consider x = -6. We substitute this value into the equation y = -2/3 x + 7, resulting in y = -2/3 * (-6) + 7. Multiplying -2/3 by -6 gives us 4. Therefore, the equation simplifies to y = 4 + 7. Adding these, we find that y = 11. Hence, when x is -6, y is 11. This step reinforces the method of evaluating functions and demonstrates the consistency of the process across different input values. The substitution of x = -6 into the equation follows the same principles as the previous calculation, ensuring accuracy and clarity in each step. The arithmetic involved, once again, is straightforward, but the importance of meticulous execution cannot be overstated. A single error in multiplication or addition can lead to an incorrect y-value, which would misrepresent the function's behavior at that point. The resulting coordinate pair (-6, 11) provides another point on the function's graph, adding to our understanding of its overall shape and direction. As we continue to evaluate the function for the remaining values in the domain, the pattern of substitution and calculation will become increasingly familiar, making the process more efficient and less prone to errors. This iterative approach not only solidifies the algebraic skills required for function evaluation but also enhances our ability to visualize and interpret the function's characteristics. Each calculated point serves as a data point, contributing to a more complete picture of the function's behavior and its relationship between input and output values. In the context of linear functions, these points will ultimately align along a straight line, illustrating the constant rate of change that defines linear relationships. The ability to accurately evaluate functions for various input values is a fundamental skill in mathematics and its applications, and each calculation reinforces this crucial ability.

H2: Evaluating y when x = 3

Now, let’s evaluate the function when x = 3. Substituting x = 3 into the equation y = -2/3 x + 7, we get y = -2/3 * (3) + 7. Multiplying -2/3 by 3 gives us -2. So, the equation becomes y = -2 + 7. Adding these numbers, we find that y = 5. Thus, when x is 3, y is 5. This evaluation further demonstrates the process of substituting positive values into the function and handling negative fractions effectively. The multiplication of -2/3 and 3 requires careful attention to the signs, ensuring that the resulting product is negative. The subsequent addition of -2 and 7 follows the rules of integer arithmetic, leading to the final y-value of 5. The coordinate pair (3, 5) now represents another point on the function's graph, contributing to our growing understanding of its behavior. As we continue to evaluate the function for additional values, we are essentially mapping out its trajectory in the coordinate plane. This point-by-point approach is particularly useful for visualizing the function's shape and identifying any key features, such as intercepts or turning points. In the case of linear functions, three points are generally sufficient to define the line, but evaluating more points can provide additional confidence in the accuracy of our calculations and the overall representation of the function. The ability to accurately evaluate functions for both positive and negative values is a crucial skill in mathematics and its applications. It allows us to explore the full range of the function's behavior and to make predictions about its output for any given input. The consistent application of the substitution and calculation process reinforces this skill and enhances our overall mathematical proficiency.

H2: Evaluating y when x = 15

Finally, we evaluate the function for x = 15. Substituting x = 15 into y = -2/3 x + 7, we have y = -2/3 * (15) + 7. Multiplying -2/3 by 15 yields -10. So, the equation becomes y = -10 + 7. Adding these, we find y = -3. Therefore, when x is 15, y is -3. This final evaluation completes the table and gives us a comprehensive understanding of how the function behaves within the given domain. The multiplication of -2/3 and 15, resulting in -10, demonstrates the importance of handling fractions and negative numbers with precision. The subsequent addition of -10 and 7, yielding -3, reinforces the rules of integer arithmetic. The resulting coordinate pair (15, -3) provides the last point needed to define the function's behavior within the specified domain. With all four points now calculated, we have a clear picture of how the function maps each x-value to its corresponding y-value. This complete set of evaluations allows us to confidently populate the table and to visualize the function's graph, which in this case is a straight line. The ability to accurately evaluate functions for a variety of input values is a cornerstone of mathematical analysis and problem-solving. It enables us to model real-world phenomena, make predictions, and gain insights into the relationships between variables. By consistently applying the principles of substitution and calculation, we strengthen our mathematical skills and enhance our ability to work with functions effectively.

Completing the Table

H2: The Completed Table

Now that we've calculated the y-values for each x-value in the domain, we can complete the table:

This table visually represents the function's behavior over the given domain. This completed table provides a concise and organized summary of the function's behavior within the specified domain. Each row of the table represents a coordinate pair ( x, y ) that lies on the graph of the function. These points can be plotted on a coordinate plane to visualize the linear relationship defined by the equation y = -2/3 x + 7. The table also highlights the consistent application of the function's rule across different input values. For each x-value in the domain, the corresponding y-value is calculated using the same formula, demonstrating the predictable and deterministic nature of functions. This tabular representation is a valuable tool for understanding and communicating mathematical relationships. It allows us to quickly identify the output of the function for any given input within the domain and to observe any patterns or trends in the data. In the case of linear functions, the table will show a constant rate of change in y as x varies, reflecting the constant slope of the line. The ability to create and interpret tables of values is a fundamental skill in mathematics and its applications. It enables us to organize data, identify relationships, and make predictions about the behavior of functions. This skill is particularly useful in fields such as science, engineering, and economics, where mathematical models are used to represent and analyze real-world phenomena. The completed table serves as a tangible representation of the function's behavior and reinforces our understanding of the relationship between input and output values.

Conclusion

H2: Key Takeaways

Evaluating functions over a given domain is a fundamental skill in mathematics. This exercise not only reinforces algebraic manipulation but also deepens the understanding of function behavior. By substituting values from the domain into the function's equation, we can determine the corresponding y-values and gain insights into the function's graph and properties. This process is essential for solving mathematical problems, modeling real-world scenarios, and developing a strong foundation in mathematics. The ability to accurately evaluate functions is crucial for success in higher-level math courses and in various scientific and technical fields. The understanding of functions and their domains is a cornerstone of mathematical literacy and is essential for anyone seeking to apply mathematical principles to practical problems. The consistent practice of evaluating functions for specific input values strengthens our algebraic skills and enhances our ability to interpret and analyze mathematical relationships. The completed table serves as a valuable tool for visualizing the function's behavior and for communicating mathematical information effectively. In conclusion, mastering the process of function evaluation is a critical step in developing a comprehensive understanding of mathematics and its applications. This skill empowers us to solve problems, make predictions, and gain insights into the world around us. The ability to work with functions confidently and accurately is a testament to our mathematical proficiency and opens doors to further exploration and discovery in the field of mathematics and beyond. The importance of functions extends beyond the classroom, as they are used extensively in various disciplines to model and analyze complex systems. From physics and engineering to economics and computer science, functions provide a powerful framework for understanding and predicting the behavior of the world around us. By developing a strong foundation in function evaluation, we equip ourselves with a valuable tool for success in a wide range of endeavors.