Evaluating The Function F(x) = 2x³ - 3x² + 7 A Step-by-Step Guide

Jul 16, 2025 by ADMIN 66 views

Evaluating the Function f(x) = 2x³ - 3x² + 7: A Comprehensive Guide

#H1 Introduction

In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding various phenomena. Among these functions, polynomial functions hold a prominent position due to their versatility and wide range of applications. This comprehensive guide delves into the intricacies of evaluating a specific polynomial function, f(x) = 2x³ - 3x² + 7, at different values of x. We will meticulously walk through the step-by-step process of substituting the given values into the function and simplifying the expression to obtain the corresponding output. By the end of this exploration, you will have a firm grasp of how to evaluate polynomial functions and apply this knowledge to solve a variety of mathematical problems. Polynomial functions, like the one we are exploring, f(x) = 2x³ - 3x² + 7, are essential in modeling real-world scenarios, from projectile motion in physics to economic growth in finance. The ability to evaluate these functions at specific points is a cornerstone of mathematical understanding and problem-solving. Throughout this guide, we will emphasize clarity and precision, ensuring that each step is explained in detail. This approach will not only help you understand the mechanics of function evaluation but also foster a deeper appreciation for the underlying mathematical concepts. As we progress, we will highlight common pitfalls and provide strategies to avoid them, making this guide a valuable resource for both beginners and those seeking to refine their skills. We will also touch upon the significance of function evaluation in various fields, showcasing the practical relevance of this mathematical operation. Our goal is to transform the seemingly simple task of evaluating a function into an enriching learning experience. By engaging with this guide, you will not only learn how to solve specific problems but also develop a broader understanding of mathematical principles. So, let's embark on this journey of mathematical discovery and unlock the power of function evaluation. The function f(x) = 2x³ - 3x² + 7 is a cubic polynomial, which means it has a degree of 3. The degree of a polynomial is the highest power of the variable (in this case, x). Polynomial functions are characterized by their smooth, continuous graphs and their ability to model a wide variety of relationships. Evaluating a polynomial function involves substituting a specific value for the variable and performing the arithmetic operations to find the corresponding function value. This process is fundamental in mathematics and has applications in various fields, including engineering, physics, and computer science. Understanding how to evaluate functions accurately is crucial for solving equations, graphing functions, and making predictions based on mathematical models. In the following sections, we will apply this knowledge to evaluate f(x) = 2x³ - 3x² + 7 at different values of x. This exercise will not only reinforce the mechanics of function evaluation but also provide insights into the behavior of polynomial functions. We will explore how the function's output changes as the input varies, which is a key aspect of understanding functions in general.

H2 Evaluating f(x) at x = -1

To begin our exploration, let's evaluate the function f(x) = 2x³ - 3x² + 7 at x = -1. This involves substituting -1 for x in the function's expression and simplifying the resulting arithmetic expression. This process is a fundamental skill in algebra and is essential for understanding how functions behave. When we substitute x = -1 into the function, we get f(-1) = 2(-1)³ - 3(-1)² + 7. The next step is to evaluate the powers of -1. Remember that (-1)³ = -1 * -1 * -1 = -1, and (-1)² = -1 * -1 = 1. These are crucial steps in simplifying the expression. Now, we can substitute these values back into the equation: f(-1) = 2(-1) - 3(1) + 7. Next, we perform the multiplications: 2(-1) = -2 and 3(1) = 3. So, the expression becomes f(-1) = -2 - 3 + 7. Finally, we perform the additions and subtractions from left to right: -2 - 3 = -5, and -5 + 7 = 2. Therefore, f(-1) = 2. This result tells us that when the input to the function is -1, the output is 2. This point (-1, 2) lies on the graph of the function. Understanding this process is key to analyzing the function's behavior and its graph. Evaluating functions at different points allows us to map out the function's curve and identify key features such as intercepts and turning points. The substitution of x = -1 is a specific instance of a more general technique. We can substitute any real number for x and obtain a corresponding value for f(x). This versatility makes functions powerful tools for modeling and solving problems in various fields. The process of evaluating f(x) = 2x³ - 3x² + 7 at x = -1 highlights the importance of order of operations (PEMDAS/BODMAS). We first evaluated the exponents, then performed the multiplications, and finally the additions and subtractions. Following the correct order of operations is crucial for obtaining the correct result. A common mistake is to perform the additions and subtractions before the multiplications, which would lead to an incorrect answer. Therefore, careful attention to the order of operations is paramount. The result f(-1) = 2 provides valuable information about the function's behavior. It tells us that the function has a point at (-1, 2) on its graph. This point can be used in conjunction with other points to sketch the function's curve. Furthermore, the value of f(-1) can be used to solve equations involving the function. For example, if we were asked to find the value of x for which f(x) = 2, we would know that x = -1 is one solution. This demonstrates the practical utility of function evaluation in solving mathematical problems. In summary, evaluating f(x) = 2x³ - 3x² + 7 at x = -1 involves substituting -1 for x, simplifying the expression using the order of operations, and obtaining the result f(-1) = 2. This process is a fundamental skill in algebra and provides valuable insights into the function's behavior. The ability to evaluate functions accurately is essential for solving equations, graphing functions, and making predictions based on mathematical models.

H2 Evaluating f(x) at x = 1

Now, let's evaluate the function f(x) = 2x³ - 3x² + 7 at x = 1. This follows the same principle as the previous example, but with a different input value. Substituting x = 1 into the function, we get f(1) = 2(1)³ - 3(1)² + 7. This substitution is the first crucial step in the evaluation process. Next, we need to evaluate the powers of 1. Since 1 raised to any power is 1, we have (1)³ = 1 and (1)² = 1. This simplifies the expression significantly. Substituting these values back into the equation, we get f(1) = 2(1) - 3(1) + 7. Now, we perform the multiplications: 2(1) = 2 and 3(1) = 3. The expression becomes f(1) = 2 - 3 + 7. Finally, we perform the additions and subtractions from left to right: 2 - 3 = -1, and -1 + 7 = 6. Therefore, f(1) = 6. This result tells us that when the input to the function is 1, the output is 6. This point (1, 6) lies on the graph of the function. Understanding this evaluation helps us visualize the function's behavior around x = 1. The process of evaluating f(x) = 2x³ - 3x² + 7 at x = 1 reinforces the importance of careful substitution and simplification. Each step must be performed accurately to arrive at the correct result. A common mistake is to overlook the order of operations or to make errors in arithmetic. By practicing these evaluations, we can improve our skills and avoid such mistakes. The result f(1) = 6 provides another data point for understanding the function's graph. By plotting this point along with other evaluated points, we can begin to see the overall shape of the curve. This visual representation is a powerful tool for analyzing the function's behavior and identifying key features. The evaluation at x = 1 also highlights the role of the constant term in the function. The constant term, 7, shifts the entire graph vertically. Without this term, the graph would pass through the origin (0, 0). The presence of the constant term makes the function more versatile and allows it to model a wider range of relationships. In summary, evaluating f(x) = 2x³ - 3x² + 7 at x = 1 involves substituting 1 for x, simplifying the expression using the order of operations, and obtaining the result f(1) = 6. This process is a fundamental skill in algebra and provides valuable insights into the function's behavior. The ability to evaluate functions accurately is essential for solving equations, graphing functions, and making predictions based on mathematical models. The systematic approach we have used in this evaluation can be applied to any polynomial function and any input value. This makes it a valuable tool for mathematical problem-solving. By mastering this technique, you will be well-equipped to tackle more complex mathematical challenges.

H2 Evaluating f(x) at x = 2

Now, let's evaluate the function f(x) = 2x³ - 3x² + 7 at x = 2. This will further illustrate the process of function evaluation and provide additional insights into the function's behavior. Substituting x = 2 into the function, we get f(2) = 2(2)³ - 3(2)² + 7. This substitution is the first crucial step in the evaluation process. Next, we need to evaluate the powers of 2. We have (2)³ = 2 * 2 * 2 = 8 and (2)² = 2 * 2 = 4. These calculations are essential for simplifying the expression. Substituting these values back into the equation, we get f(2) = 2(8) - 3(4) + 7. Now, we perform the multiplications: 2(8) = 16 and 3(4) = 12. The expression becomes f(2) = 16 - 12 + 7. Finally, we perform the additions and subtractions from left to right: 16 - 12 = 4, and 4 + 7 = 11. Therefore, f(2) = 11. This result tells us that when the input to the function is 2, the output is 11. This point (2, 11) lies on the graph of the function. This evaluation helps us understand how the function's value changes as x increases. The process of evaluating f(x) = 2x³ - 3x² + 7 at x = 2 demonstrates the importance of careful calculation and attention to detail. Each step must be performed accurately to arrive at the correct result. A common mistake is to miscalculate the powers or to make errors in the arithmetic operations. By practicing these evaluations, we can improve our skills and avoid such mistakes. The result f(2) = 11 provides yet another data point for understanding the function's graph. By plotting this point along with the previously evaluated points, we can get a clearer picture of the curve's shape. This visual representation is invaluable for analyzing the function's behavior and identifying key features such as increasing and decreasing intervals. The evaluation at x = 2 also highlights the contribution of each term in the polynomial to the overall function value. The cubic term (2x³) and the quadratic term (-3x²) both influence the function's behavior, and the constant term (7) shifts the graph vertically. Understanding the role of each term is crucial for analyzing and interpreting polynomial functions. In summary, evaluating f(x) = 2x³ - 3x² + 7 at x = 2 involves substituting 2 for x, simplifying the expression using the order of operations, and obtaining the result f(2) = 11. This process is a fundamental skill in algebra and provides valuable insights into the function's behavior. The ability to evaluate functions accurately is essential for solving equations, graphing functions, and making predictions based on mathematical models. The systematic approach we have used in this evaluation can be applied to any polynomial function and any input value. This makes it a valuable tool for mathematical problem-solving. By mastering this technique, you will be well-equipped to tackle more complex mathematical challenges and to apply functions to real-world problems.

H2 Summary of Results

In the previous sections, we have meticulously evaluated the function f(x) = 2x³ - 3x² + 7 at three different values of x: -1, 1, and 2. Let's summarize our findings:

f(-1) = 2
f(1) = 6
f(2) = 11

These results provide us with three points on the graph of the function: (-1, 2), (1, 6), and (2, 11). These points can be plotted on a coordinate plane to begin visualizing the function's curve. Furthermore, these values can be used to solve equations involving the function or to make predictions based on the function's behavior. The summary of results underscores the importance of accurate function evaluation. Each value we obtained represents a specific point on the function's graph and provides valuable information about the function's behavior. A mistake in the evaluation process would lead to an incorrect point and could significantly alter our understanding of the function. Therefore, it is essential to practice and refine our function evaluation skills. The results also highlight the variability of polynomial functions. As we changed the input value (x), the output value (f(x)) changed accordingly. This variability is what makes functions so useful for modeling real-world phenomena. By understanding how the output changes in response to changes in the input, we can gain valuable insights into the relationships being modeled. In the case of f(x) = 2x³ - 3x² + 7, the function's value increases as x increases from -1 to 2. This indicates that the function is increasing over this interval. However, to get a complete picture of the function's behavior, we would need to evaluate it at additional points and consider its derivative. The summary of results also provides a foundation for further analysis of the function. We can use these values to estimate the function's slope between different points, to identify potential turning points, and to make predictions about the function's behavior outside the interval we have considered. For example, we can see that the function's value increases more rapidly between x = 1 and x = 2 than it does between x = -1 and x = 1. This suggests that the function's slope is greater in the interval [1, 2] than in the interval [-1, 1]. In conclusion, the summary of results provides a concise overview of our function evaluation efforts and highlights the importance of accurate calculations, the variability of polynomial functions, and the foundation for further analysis. These results can be used to graph the function, solve equations, and make predictions based on the function's behavior. By mastering the technique of function evaluation, you will be well-equipped to tackle a wide range of mathematical problems.

In this comprehensive guide, we have thoroughly explored the process of evaluating the function f(x) = 2x³ - 3x² + 7 at various values of x. We systematically substituted the given values into the function's expression, carefully simplified the resulting arithmetic expressions, and obtained the corresponding output values. This step-by-step approach has not only demonstrated the mechanics of function evaluation but also highlighted the importance of accuracy, attention to detail, and the order of operations. By mastering these skills, you will be well-equipped to tackle a wide range of mathematical problems involving functions. We have seen how evaluating a function at different points provides valuable insights into its behavior. By plotting the points we obtained, we can begin to visualize the function's graph and identify key features such as increasing and decreasing intervals, turning points, and intercepts. This visual representation is a powerful tool for understanding and analyzing functions. Furthermore, we have emphasized the practical relevance of function evaluation in various fields. Functions are used to model real-world phenomena in engineering, physics, economics, and many other disciplines. The ability to evaluate functions accurately is essential for making predictions, solving problems, and gaining insights from these models. The function f(x) = 2x³ - 3x² + 7 is a specific example of a polynomial function. Polynomial functions are a fundamental class of functions in mathematics, and they have a wide range of applications. By understanding how to evaluate polynomial functions, you will be well-prepared to work with more complex mathematical models. The techniques we have discussed in this guide can be applied to any function, not just polynomial functions. The general principle of substituting a value for the variable and simplifying the expression applies to trigonometric functions, exponential functions, logarithmic functions, and many others. Therefore, the skills you have learned in this guide will be valuable throughout your mathematical journey. As you continue to study mathematics, you will encounter increasingly complex functions and mathematical models. However, the fundamental skill of function evaluation will remain essential. By practicing and refining this skill, you will build a strong foundation for future success. In conclusion, we have achieved our goal of providing a comprehensive guide to evaluating the function f(x) = 2x³ - 3x² + 7. We have demonstrated the process step-by-step, highlighted the importance of accuracy and attention to detail, and emphasized the practical relevance of function evaluation. By mastering these skills, you will be well-equipped to tackle a wide range of mathematical problems and to apply functions to real-world situations. We encourage you to continue practicing and exploring functions to deepen your understanding and to unlock the power of mathematics.