Completing The Table For F(x) = X^5 + (x + 3)^2
Completing tables based on functions is a fundamental concept in mathematics, bridging the gap between abstract equations and concrete numerical values. In this article, we will delve into the process of evaluating a function and using the results to fill in a table. Our specific focus will be on the function f(x) = x^5 + (x + 3)^2, and we'll determine the missing value in a given table. This exploration will reinforce the understanding of function evaluation and its practical applications. By carefully examining each component of the function and meticulously performing the calculations, we will arrive at the correct solution and enhance our comprehension of function behavior. Understanding how to evaluate functions and complete tables is crucial for various mathematical applications, including graphing, data analysis, and problem-solving. This article will provide a comprehensive guide to this process, ensuring clarity and accuracy in each step.
Understanding the Function f(x) = x^5 + (x + 3)^2
To accurately complete the table, the first crucial step is to fully grasp the function itself. The function f(x) = x^5 + (x + 3)^2 is a polynomial function, comprising two primary terms: x^5 and (x + 3)^2. Let's dissect each term individually to understand their contributions to the overall function behavior.
The first term, x^5, represents x raised to the power of 5. This term will exhibit significant changes as the value of x varies, especially for values farther from zero. When x is positive, x^5 will also be positive, and when x is negative, x^5 will be negative. The magnitude of x^5 will increase rapidly as the absolute value of x grows, reflecting the nature of a quintic function. For instance, if x equals 2, then x^5 equals 32, and if x equals -2, then x^5 equals -32. This rapid growth makes x^5 a dominant term in the function for larger values of x. The behavior of x^5 is symmetrical about the origin, meaning that the function's value at x and -x will have opposite signs but similar magnitudes.
The second term, (x + 3)^2, represents the square of the binomial (x + 3). This term is a quadratic expression, which means it will produce a parabolic shape when graphed. Squaring any real number results in a non-negative value, so (x + 3)^2 will always be greater than or equal to zero. The minimum value of this term occurs when x equals -3, making the entire expression equal to zero. For values of x greater or smaller than -3, the term (x + 3)^2 will increase. This quadratic term contributes to the function's overall shape by adding a squared component, which affects the rate of change and introduces a curve. The addition of 3 inside the parenthesis shifts the parabola horizontally, with the vertex of the parabola located at x = -3.
By understanding these individual terms, we can better anticipate the function's behavior and accurately calculate its values for different inputs. The combination of the quintic term x^5 and the quadratic term (x + 3)^2 creates a unique function with a blend of rapid growth and parabolic influence. Evaluating the function requires careful attention to the order of operations, ensuring that the exponentiation and squaring are performed before addition. This thorough understanding of the function’s components is essential for correctly completing the table and solving related problems.
The Given Table and the Missing Value
The table provided presents a set of x values and their corresponding f(x) values, derived from the function f(x) = x^5 + (x + 3)^2. The table serves as a snapshot of the function’s behavior at specific points, allowing us to observe how the output f(x) changes as the input x varies. This tabular representation is a practical way to visualize discrete function values and identify patterns or trends. Understanding how to interpret and complete such tables is a critical skill in mathematics, especially when dealing with functions and their applications.
The table includes the following pairs of (x, f(x)) values:
- When x = -2, f(x) = -31
- When x = -1, f(x) = ? (This is the missing value we need to find)
- When x = 0, f(x) = 9
- When x = 1, f(x) = 17
Each row in the table represents a specific evaluation of the function at the given x value. For instance, the first row tells us that when x is -2, the function f(x) evaluates to -31. This means that when -2 is plugged into the equation f(x) = x^5 + (x + 3)^2, the result of the calculation is -31. This serves as a verification point and helps us understand the function’s output for negative inputs.
Similarly, the row for x = 0 shows that f(0) = 9. This indicates that when x is 0, the function value is 9. Plugging 0 into the function, we get f(0) = 0^5 + (0 + 3)^2 = 0 + 3^2 = 9, which confirms the table's entry. This value is particularly useful because it often simplifies calculations due to the presence of zero in the terms.
The row for x = 1 shows that f(1) = 17. This means that when x is 1, the function value is 17. Substituting 1 into the function, we have f(1) = 1^5 + (1 + 3)^2 = 1 + 4^2 = 1 + 16 = 17, which again confirms the table's entry. This value helps us understand the function's behavior for positive inputs.
The central task now is to determine the missing value when x = -1. This requires us to substitute -1 into the function f(x) = x^5 + (x + 3)^2 and perform the calculation. Finding this missing value will complete the table and provide a more comprehensive understanding of the function's behavior across the given range of x values. The process involves careful evaluation of each term in the function, ensuring we follow the correct order of operations to arrive at the accurate result.
Calculating the Missing Value
To find the missing value in the table, 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 and performing the necessary calculations. By carefully following the order of operations, we can determine the correct value of f(-1) and complete the table.
The first step is to replace x with -1 in the function: f(-1) = (-1)^5 + (-1 + 3)^2. This substitution sets up the expression we need to evaluate. Now, we will address each term separately, starting with (-1)^5.
The term (-1)^5 represents -1 raised to the power of 5. Since 5 is an odd exponent, the result will be negative. Calculating this, we have (-1)^5 = -1. This is because -1 multiplied by itself an odd number of times yields -1. Understanding this behavior of negative numbers raised to odd powers is crucial for accurate calculations.
Next, we evaluate the second term, (-1 + 3)^2. First, we need to simplify the expression inside the parentheses: (-1 + 3) = 2. Now, we have 2^2, which means 2 squared. Calculating this, we get 2^2 = 4. Squaring a number means multiplying it by itself, so 2 squared is 2 times 2, which equals 4.
Now that we have evaluated both terms, we can combine them to find f(-1). The expression becomes f(-1) = -1 + 4. Adding these values together, we get f(-1) = 3. This is the missing value we were looking for.
Therefore, when x = -1, the function f(x) = x^5 + (x + 3)^2 evaluates to 3. This completes the table, providing a full set of x and f(x) values for the given function. This calculation highlights the importance of correctly applying the order of operations and understanding the properties of exponents and negative numbers. With the missing value determined, we can now present the completed table, giving a clear picture of the function’s behavior at these specific points.
The Completed Table and the Correct Answer
Having calculated the missing value for f(x) when x = -1, we can now complete the table. The completed table provides a clear representation of the function f(x) = x^5 + (x + 3)^2 at the given x values. This complete set of data points allows for a better understanding of the function's behavior within the specified domain.
Here is the completed table:
| x | f(x) |
|---|---|
| -2 | -31 |
| -1 | 3 |
| 0 | 9 |
| 1 | 17 |
The missing value, which we calculated to be 3, now fills the gap in the table. This value indicates that when x is -1, the function f(x) evaluates to 3. With the table now complete, we can see a progression of f(x) values as x increases from -2 to 1. This progression gives us a glimpse into the function’s overall behavior, showing how the output changes in response to different inputs.
Looking back at the original question, we were asked to identify the value that completes the table. Among the options provided:
- A. -17
- B. -3
- C. 1
- D. 3
We have determined that the correct value is 3, which corresponds to option D. Therefore, the answer to the question is D. 3. This conclusion is based on our step-by-step evaluation of the function f(x) = x^5 + (x + 3)^2 at x = -1, which yielded the result of 3.
This exercise of completing the table reinforces the importance of accurate function evaluation and attention to detail in mathematical calculations. By systematically substituting values into the function and following the correct order of operations, we were able to successfully determine the missing value and complete the table. This process not only answers the specific question but also enhances our understanding of how functions behave and how they can be represented in tabular form. The completed table serves as a valuable tool for visualizing and analyzing the function’s characteristics within the given interval of x values.
Conclusion
In summary, we have successfully completed the table for the function f(x) = x^5 + (x + 3)^2 by accurately calculating the missing value. The process involved understanding the function's components, substituting the given x value into the function, and following the correct order of operations to arrive at the result. The missing value was determined to be 3, which completes the table as follows:
| x | f(x) |
|---|---|
| -2 | -31 |
| -1 | 3 |
| 0 | 9 |
| 1 | 17 |
This exercise underscores the importance of function evaluation in mathematics. Being able to substitute values into a function and calculate the corresponding output is a fundamental skill that has wide-ranging applications. From graphing functions to solving equations and analyzing data, function evaluation is a key tool in mathematical problem-solving.
Furthermore, this article highlights the significance of attention to detail and systematic calculation. The function f(x) = x^5 + (x + 3)^2 involves multiple terms and operations, including exponentiation and addition. To correctly evaluate the function, it is crucial to follow the order of operations (PEMDAS/BODMAS) and pay close attention to signs, particularly when dealing with negative numbers. A small error in calculation can lead to an incorrect result, emphasizing the need for careful and methodical work.
The ability to complete tables based on functions is not only a mathematical skill but also a valuable tool for visualizing and understanding functional relationships. Tables provide a discrete set of data points that can help us see patterns and trends in a function's behavior. They can also serve as a basis for graphing the function, allowing for a visual representation of the relationship between x and f(x).
In conclusion, by understanding the function, applying the correct calculations, and completing the table, we have demonstrated a comprehensive approach to solving mathematical problems involving function evaluation. This process reinforces essential mathematical skills and provides a foundation for more advanced topics in algebra and calculus. The correct answer to the question is D. 3, which completes the table and provides a clear picture of the function’s behavior at the given points.
