Finding The Function Rule That Models A Table Of Values

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A. $f(x)=3x+10$ B. $f(x)=2x+3$ C. Discussion category: mathematics

Unveiling the Function Rule: A Step-by-Step Guide

When faced with the task of identifying the function rule that models a given set of data, as presented in the table above, it's crucial to adopt a systematic approach. This involves not only understanding the fundamental concepts of functions but also employing various techniques to analyze the provided data points and deduce the underlying relationship between the input (x) and output (f(x)) values. In this comprehensive guide, we will dissect the problem, explore different methods for finding the function rule, and ultimately arrive at the correct solution. Our key objective is to determine the function f(x) that accurately maps the given x-values to their corresponding f(x)-values. The provided table offers a glimpse into the function's behavior, and our goal is to extrapolate this behavior into a generalized rule. To achieve this, we'll delve into the realm of linear functions, scrutinize the data for patterns, and employ algebraic techniques to pinpoint the function rule that perfectly encapsulates the relationship between x and f(x).

Our initial step involves examining the nature of the function. Is it linear, quadratic, exponential, or something else entirely? A linear function exhibits a constant rate of change, meaning that for every consistent increment in x, there is a corresponding consistent increment in f(x). This characteristic is readily identifiable by observing the differences between successive f(x)-values for equal intervals of x. In the given table, we can see that as x increases by 4 (from -7 to -1), f(x) increases by 12 (from -11 to 1). Similarly, as x increases by 4 (from -1 to 3), f(x) increases by 8 (from 1 to 9). This observation hints at a linear relationship, but we need to confirm this across all data points. The constant rate of change, also known as the slope, is a critical indicator of linearity. To calculate the slope, we can use the formula: slope (m) = (change in f(x)) / (change in x). Applying this to our data, we get m = (1 - (-11)) / (-1 - (-7)) = 12 / 6 = 2. This calculation provides strong evidence that the function is indeed linear. Now that we have established the linearity of the function, we can proceed to determine the specific equation that represents it. The general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept (the value of f(x) when x is 0). We have already calculated the slope (m) to be 2. The next step is to find the y-intercept (b). To do this, we can substitute one of the data points from the table into the equation f(x) = 2x + b and solve for b. Let's use the point (-1, 1). Plugging these values into the equation, we get 1 = 2(-1) + b. Solving for b, we have 1 = -2 + b, which gives us b = 3. Therefore, the linear function that models the data is f(x) = 2x + 3. This function rule suggests that for any given x-value, we can multiply it by 2 and add 3 to obtain the corresponding f(x)-value.

Method 1: Spotting the Pattern - A Detailed Exploration

One of the most intuitive methods for determining a function rule is to meticulously examine the data presented in the table and identify any discernible patterns. This approach relies heavily on observation skills and the ability to discern relationships between the input (x) and output (f(x)) values. In the given table, we have a set of ordered pairs: (-7, -11), (-1, 1), (3, 9), (4, 11), and (7, 17). Our mission is to uncover the mathematical rule that connects these pairs. The key principle here is to look for consistent changes or relationships. For instance, we can analyze how the f(x) values change as the x values change. A systematic approach is paramount. We begin by observing the differences between consecutive f(x) values. The difference between 1 and -11 is 12. The difference between 9 and 1 is 8. The difference between 11 and 9 is 2. The difference between 17 and 11 is 6. These differences are not constant, which might initially suggest that the function is not linear. However, it's crucial to consider the corresponding changes in the x values. The difference between -1 and -7 is 6. The difference between 3 and -1 is 4. The difference between 4 and 3 is 1. The difference between 7 and 4 is 3. To determine if the function is linear, we need to check if the ratio of the change in f(x) to the change in x is constant. This ratio represents the slope of the line. Let's calculate these ratios: (1 - (-11)) / (-1 - (-7)) = 12 / 6 = 2 (9 - 1) / (3 - (-1)) = 8 / 4 = 2 (11 - 9) / (4 - 3) = 2 / 1 = 2 (17 - 11) / (7 - 4) = 6 / 3 = 2 The consistent ratio of 2 strongly indicates that the function is linear. This is a crucial finding as it narrows down our search to linear function rules of the form f(x) = mx + b, where m is the slope and b is the y-intercept. We have already determined that the slope (m) is 2. Now, we need to find the y-intercept (b). This is the value of f(x) when x is 0. Unfortunately, our table does not directly provide the value of f(0). However, we can use one of the data points and the slope to calculate b. Let's use the point (-1, 1). Substituting x = -1 and f(x) = 1 into the equation f(x) = 2x + b, we get: 1 = 2(-1) + b 1 = -2 + b b = 3 Therefore, the y-intercept (b) is 3. Combining the slope (m = 2) and the y-intercept (b = 3), we arrive at the function rule: f(x) = 2x + 3. This rule suggests that for any given x-value, we multiply it by 2 and add 3 to obtain the corresponding f(x)-value. To verify this rule, we can test it against all the data points in the table. For example, when x = -7, f(x) = 2(-7) + 3 = -14 + 3 = -11, which matches the table. When x = -1, f(x) = 2(-1) + 3 = -2 + 3 = 1, which also matches the table. Continuing this process for all data points will confirm the accuracy of the function rule. By meticulously spotting the pattern and systematically analyzing the data, we have successfully identified the function rule that models the given data.

Method 2: Leveraging the Slope-Intercept Form - A Detailed Explanation

The slope-intercept form of a linear equation, f(x) = mx + b, provides a powerful framework for determining the function rule. This method is particularly effective when dealing with linear functions, as it directly relates the slope (m) and y-intercept (b) to the function's behavior. Our primary goal here is to calculate the slope and y-intercept using the given data points and then construct the function rule. The slope (m) represents the rate of change of the function, indicating how much f(x) changes for each unit change in x. To calculate the slope, we need two distinct points from the table. Let's choose the points (-7, -11) and (-1, 1). Using the slope formula, m = (f(x2) - f(x1)) / (x2 - x1), we get: m = (1 - (-11)) / (-1 - (-7)) m = 12 / 6 m = 2 This calculation confirms that the slope of the line is 2. This is a crucial piece of information as it gives us the coefficient of x in the function rule. Now that we have the slope, we need to determine the y-intercept (b). The y-intercept is the point where the line crosses the y-axis, which occurs when x = 0. Our table does not directly provide the value of f(0), but we can calculate it using the slope and one of the data points. We can substitute the slope (m = 2) and the coordinates of one of the points into the slope-intercept form (f(x) = mx + b) and solve for b. Let's use the point (-1, 1). Substituting x = -1 and f(x) = 1, we get: 1 = 2(-1) + b 1 = -2 + b b = 3 Therefore, the y-intercept (b) is 3. This value represents the constant term in the function rule. Having determined both the slope (m = 2) and the y-intercept (b = 3), we can now construct the function rule in slope-intercept form: f(x) = 2x + 3. This function rule suggests that for any given x-value, we multiply it by 2 and add 3 to obtain the corresponding f(x)-value. To ensure the accuracy of this function rule, it's essential to verify it against all the data points in the table. For instance, when x = 3, f(x) = 2(3) + 3 = 6 + 3 = 9, which matches the table. When x = 4, f(x) = 2(4) + 3 = 8 + 3 = 11, which also matches the table. This verification process confirms that the function rule accurately models the given data. The slope-intercept form provides a direct and efficient method for determining the function rule, especially for linear functions. By calculating the slope and y-intercept, we can readily construct the equation that represents the relationship between x and f(x).

Method 3: Utilizing Point-Slope Form - A Detailed Explanation

The point-slope form of a linear equation, f(x) - f(x1) = m(x - x1), offers another effective method for determining the function rule. This form is particularly useful when we know the slope (m) and a point (x1, f(x1)) on the line. Our core strategy here is to calculate the slope and then use the point-slope form to derive the function rule. As with the slope-intercept method, the first step is to calculate the slope (m). We can use any two distinct points from the table. Let's choose the points (-7, -11) and (-1, 1). Applying the slope formula, m = (f(x2) - f(x1)) / (x2 - x1), we get: m = (1 - (-11)) / (-1 - (-7)) m = 12 / 6 m = 2 This calculation confirms that the slope of the line is 2, which is a vital parameter for defining the linear function. Now that we have the slope, we can utilize the point-slope form to construct the equation of the line. We need to choose one of the data points to substitute into the point-slope form. Let's use the point (-1, 1). Substituting m = 2, x1 = -1, and f(x1) = 1 into the point-slope form, we get: f(x) - 1 = 2(x - (-1)) f(x) - 1 = 2(x + 1) Next, we need to simplify this equation to obtain the function rule in slope-intercept form (f(x) = mx + b). Expanding the right side of the equation, we have: f(x) - 1 = 2x + 2 Adding 1 to both sides, we get: f(x) = 2x + 3 This is the function rule in slope-intercept form. It suggests that for any given x-value, we multiply it by 2 and add 3 to obtain the corresponding f(x)-value. To ensure the accuracy of this function rule, it's crucial to verify it against all the data points in the table. For example, when x = 7, f(x) = 2(7) + 3 = 14 + 3 = 17, which matches the table. When x = 3, f(x) = 2(3) + 3 = 6 + 3 = 9, which also matches the table. This verification process confirms that the function rule accurately models the given data. The point-slope form provides a flexible and efficient method for determining the function rule, particularly when the slope and a point on the line are known. By substituting the slope and the coordinates of a point into the point-slope form and simplifying, we can readily obtain the equation that represents the relationship between x and f(x).

Solution and Justification

Based on our analysis using multiple methods, the function rule that models the function over the domain specified in the table is B. f(x) = 2x + 3. We arrived at this conclusion by:

1. Spotting the Pattern: We observed a consistent rate of change in f(x) relative to x, indicating a linear relationship.
2. Leveraging the Slope-Intercept Form: We calculated the slope (m = 2) and y-intercept (b = 3) and constructed the function rule f(x) = 2x + 3.
3. Utilizing Point-Slope Form: We used the point-slope form to derive the same function rule, further confirming its validity.

Each of these methods independently led us to the same function rule, reinforcing our confidence in the solution. The other options, A. f(x) = 3x + 10, can be easily disproven by substituting the x-values from the table and observing that they do not yield the corresponding f(x)-values. For example, if we substitute x = -7 into f(x) = 3x + 10, we get f(x) = 3(-7) + 10 = -21 + 10 = -11, which matches the table. However, if we substitute x = -1, we get f(x) = 3(-1) + 10 = -3 + 10 = 7, which does not match the table. This inconsistency demonstrates that option A is incorrect. In contrast, our derived function rule, f(x) = 2x + 3, accurately maps all the given x-values to their corresponding f(x)-values, making it the correct solution.

In conclusion, understanding the fundamental concepts of functions, employing systematic methods, and verifying the results are crucial for accurately determining the function rule that models a given set of data. By mastering these techniques, you can confidently tackle similar problems and gain a deeper appreciation for the power of mathematical modeling.