Completing The Table For Function F(x) = X^5 + (x + 3)^2
This article delves into the function f(x) = x^5 + (x + 3)^2 and its application in completing a table of values. We will explore how to evaluate the function for a given input and determine the missing value in the provided table. Understanding function evaluation is a fundamental concept in mathematics, and this example provides a practical application of this skill. The ability to accurately evaluate functions is crucial for various mathematical tasks, including graphing, solving equations, and modeling real-world phenomena. In this particular case, we're tasked with finding the value of f(x) when x = -1. This involves substituting -1 into the function's expression and performing the necessary arithmetic operations. This exercise highlights the importance of following the order of operations (PEMDAS/BODMAS) to arrive at the correct answer. Moreover, this kind of problem is typical in introductory algebra and precalculus courses, serving as a foundational step for more advanced mathematical concepts. Solving problems like this helps students develop a strong understanding of function notation and how to work with algebraic expressions. It also reinforces the concept of a function as a mapping between inputs and outputs. The process of completing a table of values is also directly related to graphing functions. By calculating several f(x) values for different x values, we can create a set of points that can be plotted on a coordinate plane to visualize the function's behavior. This connection between tables and graphs is essential for understanding the graphical representation of functions. The function itself, f(x) = x^5 + (x + 3)^2, is a polynomial function. It combines a power function (x^5) with a quadratic function (the expansion of (x + 3)^2). Analyzing the different terms of the function can give us insights into its overall behavior. For instance, the x^5 term will dominate the function's behavior for very large positive and negative values of x, while the quadratic term will have a more significant impact for values of x closer to -3. This type of function analysis is important in calculus and other higher-level mathematics courses. Furthermore, the specific task of finding a missing value in a table tests a student's ability to apply function evaluation skills in a context. It requires not only the mechanical process of substituting and calculating but also the understanding of how a table represents the relationship between input and output values of a function. In conclusion, this problem is a valuable exercise in function evaluation and highlights the importance of this skill in various mathematical contexts. By understanding how to complete the table for f(x) = x^5 + (x + 3)^2, we gain a deeper understanding of function notation, algebraic manipulation, and the connection between functions, tables, and graphs.
Understanding the Problem
The problem presents a function, f(x) = x^5 + (x + 3)^2, and a partially completed table. The table shows several x values and their corresponding f(x) values. However, one f(x) value is missing when x = -1. Our task is to determine the correct value to complete the table. This problem fundamentally tests our ability to evaluate a function at a specific point. In essence, we need to substitute x = -1 into the function's expression and calculate the result. This seemingly simple process is a cornerstone of function theory and is essential for various mathematical applications. Understanding function evaluation is not just about plugging in numbers; it's about grasping the concept of a function as a mapping from one set of numbers (the domain) to another (the range). The table represents a discrete set of these mappings. Each row in the table shows an input-output pair, and our job is to find the missing pair. This understanding of functions as mappings is crucial for more advanced topics like calculus and differential equations. The given function, f(x) = x^5 + (x + 3)^2, is a polynomial function, specifically a combination of a power function and a quadratic function. Recognizing the types of functions involved can provide insights into their behavior. The x^5 term indicates that the function will have a significant change in its output as x varies, especially for larger values of x. The (x + 3)^2 term, being a squared term, will always be non-negative and will have a minimum value when x = -3. These characteristics help us understand the overall shape and trend of the function's graph, even before we plot it. Moreover, the problem format – a table with a missing value – is a common way to assess function evaluation skills. This type of question is frequently encountered in algebra and precalculus courses and is often used as a stepping stone to more complex problems. Completing tables is directly related to graphing functions. By finding several input-output pairs, we can plot these points on a coordinate plane and connect them to visualize the function's curve. The more points we have, the more accurate our graph will be. This connection between tables and graphs is a fundamental concept in mathematics. In addition to the mechanical process of substituting and calculating, this problem also requires careful attention to the order of operations (PEMDAS/BODMAS). We need to ensure that we perform the operations in the correct sequence: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). A mistake in the order of operations can lead to an incorrect answer. In summary, this problem is a valuable exercise in function evaluation, requiring a solid understanding of function notation, algebraic manipulation, and the order of operations. It serves as a fundamental building block for more advanced mathematical concepts and highlights the importance of functions as mappings between inputs and outputs.
Evaluating f(x) at x = -1
To find the missing value, we need to evaluate the function f(x) = x^5 + (x + 3)^2 when x = -1. This involves substituting -1 for x in the function's expression. This process of substitution is a core skill in algebra and is used extensively in various mathematical contexts. Understanding how to correctly substitute values into expressions is crucial for solving equations, simplifying expressions, and evaluating functions. In this specific case, the substitution step transforms the function's expression from a general formula to a specific numerical calculation. After substituting, we get f(-1) = (-1)^5 + (-1 + 3)^2. Now, we need to simplify this expression by following the order of operations (PEMDAS/BODMAS). This order dictates that we first handle parentheses, then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Adhering to the order of operations is essential to arrive at the correct answer. Failing to do so can lead to significant errors in the calculation. In our expression, the parentheses contain the term (-1 + 3). Evaluating this gives us 2, so the expression becomes f(-1) = (-1)^5 + (2)^2. Next, we need to handle the exponents. (-1)^5 means -1 raised to the power of 5, which is -1 multiplied by itself five times. Since a negative number raised to an odd power remains negative, (-1)^5 = -1. Similarly, (2)^2 means 2 raised to the power of 2, which is 2 multiplied by itself. Therefore, (2)^2 = 4. Substituting these values back into the expression, we get f(-1) = -1 + 4. Finally, we perform the addition: -1 + 4 = 3. Therefore, the value of the function f(x) when x = -1 is 3. This means that the missing value in the table is 3. The entire process of evaluating f(x) at x = -1 highlights several important mathematical concepts. It reinforces the idea of a function as a mapping, the process of substitution, and the importance of the order of operations. It also demonstrates how a seemingly complex expression can be simplified step-by-step to arrive at a numerical answer. Furthermore, this type of problem is a common example of how functions are used in mathematics to model relationships between variables. By evaluating a function at different points, we can understand how the output of the function changes as the input changes. This understanding is crucial for applications in various fields, including physics, engineering, and economics. In conclusion, evaluating f(x) = x^5 + (x + 3)^2 at x = -1 involves a series of steps that demonstrate core algebraic skills. The result, f(-1) = 3, is the missing value in the table and highlights the importance of function evaluation in mathematics.
Completing the Table
Having calculated f(-1) = 3, we can now complete the table. The original table had a missing value for f(x) when x = -1. By evaluating the function at this point, we have determined that the missing value is 3. Completing tables of values is a fundamental skill in mathematics, particularly when working with functions. It allows us to visualize the relationship between the input (x) and the output (f(x)) of a function. Each row in the table represents a pair of corresponding x and f(x) values, and by filling in the missing values, we gain a more complete picture of the function's behavior. The completed table now looks like this:
| x | f(x) |
|---|---|
| -2 | -31 |
| -1 | 3 |
| 0 | 9 |
| 1 | 17 |
This completed table provides a set of discrete points that lie on the graph of the function f(x) = x^5 + (x + 3)^2. These points can be plotted on a coordinate plane to visualize the function's curve. The more points we have, the more accurate our representation of the function's graph will be. This connection between tables and graphs is a crucial concept in mathematics. It allows us to transition between the numerical representation of a function (the table) and its visual representation (the graph). This ability to visualize functions is essential for understanding their behavior and properties. For instance, by looking at the table, we can observe how f(x) changes as x increases. We can also identify any patterns or trends in the function's output. In this particular case, we can see that f(x) increases as x increases within the range of x values provided in the table. However, this is just a limited view of the function's overall behavior. To get a more complete understanding, we would need to consider a wider range of x values and possibly plot the graph of the function. Completing the table also reinforces the concept of a function as a mapping between inputs and outputs. Each x value in the table is mapped to a corresponding f(x) value by the function's rule, which is given by the expression f(x) = x^5 + (x + 3)^2. Understanding this mapping is crucial for working with functions effectively. Furthermore, the process of completing the table highlights the importance of accurate calculation and attention to detail. A single error in evaluating the function can lead to an incorrect entry in the table, which can then affect our understanding of the function's behavior. In summary, completing the table for f(x) = x^5 + (x + 3)^2 with the value f(-1) = 3 is a valuable exercise in understanding function evaluation and its connection to tables and graphs. The completed table provides a set of points that can be used to visualize the function's behavior and reinforces the concept of a function as a mapping between inputs and outputs.
The Answer
The value that completes the table is 3. Therefore, the correct answer is D. 3.
Keywords Repair
What value completes the table for the function f(x) = x^5 + (x + 3)^2, given the provided table with missing value for x = -1?
