Completing Input-Output Tables For The Function G(x) = 3 - 8x

In mathematics, functions are fundamental building blocks that describe relationships between inputs and outputs. Understanding how to work with functions, especially completing input-output tables, is crucial for grasping mathematical concepts and their applications. This article delves into the process of completing input-output tables for a given function, using the example function g(x) = 3 - 8x. We will walk through the steps, providing clear explanations and examples to solidify your understanding. Let's embark on this journey to master the art of function evaluation and table completion.

Understanding Functions and Input-Output Tables

Before we dive into the specifics of completing the table for g(x) = 3 - 8x, let's establish a clear understanding of functions and input-output tables. At its core, a function is a rule or a relationship that assigns each input value to exactly one output value. Think of it as a mathematical machine: you feed in an input, and the machine processes it according to its rule and spits out a unique output. Input-output tables are a way to represent this relationship in a structured format.

The input is often represented by the variable x, and the output is represented by g(x) (in this case) or y in more general terms. The function rule, such as g(x) = 3 - 8x, dictates how the input x is transformed to produce the output g(x). An input-output table is a table with two columns, one for inputs (x) and one for corresponding outputs (g(x)). Completing such a table involves finding the missing input or output values based on the given function rule.

Input-output tables are not just abstract mathematical constructs; they have practical applications in various fields. For example, in computer science, functions are the backbone of programming, and understanding their behavior is essential for writing effective code. In physics, functions can describe the relationship between time and distance, or force and acceleration. In economics, functions can model supply and demand curves. Therefore, mastering the skill of working with functions and input-output tables provides a versatile tool for problem-solving in diverse domains. Understanding the concept of a function is the cornerstone of mathematical analysis and its real-world applications. A function, in essence, is a rule that maps each input value to a unique output value. This mapping can be represented in various ways: equations, graphs, tables, or even verbal descriptions. The input, often denoted by x, is the value that we feed into the function. The output, commonly represented by g(x) or y, is the result we obtain after applying the function's rule to the input. Input-output tables provide a structured way to visualize this relationship by listing pairs of input values and their corresponding output values. These tables are invaluable for understanding the function's behavior and identifying patterns in the relationship between inputs and outputs. The ability to construct and interpret input-output tables is a fundamental skill in mathematics, serving as a stepping stone to more advanced concepts such as graphing functions, solving equations, and modeling real-world phenomena. The function rule, in our case g(x) = 3 - 8x, acts as the engine that transforms the input into the output. This rule dictates the specific mathematical operations that must be performed on the input value to obtain the corresponding output value. In the context of g(x) = 3 - 8x, the rule instructs us to multiply the input x by -8 and then add the result to 3. By understanding and applying this rule, we can systematically generate output values for any given input value and vice versa. Mastering the interpretation and application of function rules is paramount for effectively working with functions and their various representations. In essence, functions are the language of mathematical relationships, and input-output tables are a crucial tool for deciphering and utilizing this language. Let's now delve deeper into the specific task of completing the table for our example function, g(x) = 3 - 8x, and explore the step-by-step process involved.

Completing the Table for g(x) = 3 - 8x

Now, let's focus on completing the table for the function g(x) = 3 - 8x. We have a table with some missing input (x) and output (g(x)) values, and our goal is to fill in the blanks. We'll tackle this by using the function rule to find the missing values, one at a time. The given table looks like this:

We have four missing values to determine. We will address each row systematically, demonstrating the process of using the function rule to find the missing input or output. The key to completing the table lies in understanding the relationship between the input x and the output g(x) as defined by the function rule. We will substitute known values into the equation g(x) = 3 - 8x and solve for the unknown value. This process will involve basic algebraic manipulations, such as addition, subtraction, multiplication, and division. Each step will be clearly explained to ensure a comprehensive understanding of the method. By working through each row of the table, we will gain proficiency in evaluating functions and completing input-output tables. This skill is not only essential for mastering mathematical concepts but also for applying these concepts to real-world problems. So, let's begin the process of unraveling the missing values in our table, one row at a time. This meticulous approach will allow us to appreciate the power and elegance of functions in describing mathematical relationships. The first row presents us with a missing input value when the output g(x) is 0. This means we need to find the value of x that, when plugged into the function g(x) = 3 - 8x, yields an output of 0. This process involves solving an algebraic equation. The second row, conversely, gives us an input value of 0 and asks us to find the corresponding output value. This is a more straightforward process of direct substitution. The third row, similar to the first, presents us with a missing input value for a given output. And the final row requires us to find the output for a known input, akin to the second row. By systematically addressing each row, we will not only complete the table but also reinforce our understanding of how inputs and outputs interact within a function's framework. So, let's embark on this step-by-step exploration, beginning with the first missing value.

Step 1: Finding x when g(x) = 0

In the first row, we are given g(x) = 0 and need to find the corresponding x value. To do this, we substitute 0 for g(x) in the function equation:

0 = 3 - 8x

Now, we solve for x. First, subtract 3 from both sides:

-3 = -8x

Next, divide both sides by -8:

x = -3 / -8

x = 3/8

So, when g(x) = 0, x = 3/8. Let's break down this process further. We started with the fundamental equation that defines our function, g(x) = 3 - 8x. The goal here was to isolate x, which means getting x by itself on one side of the equation. The first step towards this was to eliminate the constant term, 3, from the right side. We achieved this by subtracting 3 from both sides of the equation. This maintains the equality because whatever operation we perform on one side, we must also perform on the other. This gave us -3 = -8x. The next obstacle was the coefficient -8 multiplying x. To undo this multiplication, we performed the inverse operation: division. We divided both sides of the equation by -8. This isolated x and gave us the solution x = 3/8. This fraction represents the exact value of x that makes the function g(x) equal to 0. We can verify this by plugging x = 3/8 back into the original equation: g(3/8) = 3 - 8(3/8) = 3 - 3 = 0*. This confirms that our solution is correct. Understanding this process of solving for x is crucial for various mathematical applications, including finding the roots or zeros of a function, determining the x-intercept of a graph, and solving real-world problems modeled by equations. The ability to manipulate equations and isolate variables is a fundamental skill in algebra and beyond. It allows us to unlock the hidden relationships between quantities and make predictions about the behavior of systems. In the context of functions, solving for x when we know the output g(x) is a common task that arises in numerous scenarios. With this first missing value successfully determined, we move on to the next, where we are given an input value and asked to find the corresponding output.

Step 2: Finding g(x) when x = 0

In the second row, we are given x = 0 and need to find g(x). This is a more straightforward process than the previous step. We simply substitute 0 for x in the function equation:

g(0) = 3 - 8(0)

g(0) = 3 - 0

g(0) = 3

So, when x = 0, g(x) = 3. The process of finding the output g(x) when the input x is known is called evaluating the function. In this case, we are evaluating g(x) at x = 0. The substitution is the key step: we replace the variable x in the function's equation with the given value, in this case, 0. The order of operations (PEMDAS/BODMAS) is crucial for accurate evaluation. Here, we perform the multiplication before the subtraction. 8 multiplied by 0 equals 0, which simplifies the equation to g(0) = 3 - 0. Finally, 3 minus 0 equals 3, giving us the output value g(0) = 3. This value represents the point on the function's graph where the x-coordinate is 0, also known as the y-intercept. Evaluating functions at specific input values is a fundamental operation in mathematics and its applications. It allows us to determine the function's behavior at particular points, analyze trends, and make predictions. In real-world scenarios, evaluating functions can help us model and understand various phenomena. For instance, if g(x) represented the temperature at time x, evaluating g(0) would give us the temperature at the initial time. Similarly, if g(x) represented the cost of producing x items, evaluating g(0) would give us the fixed costs. The simplicity of this step, substituting and simplifying, belies its importance. It forms the basis for much more complex analysis of functions. With this second value determined, we move on to the third missing value, where we are once again asked to find the input x given the output g(x). This will require us to revisit the algebraic techniques we used in Step 1.

Step 3: Finding x when g(x) = -5

In the third row, we have g(x) = -5, and we need to find the corresponding x value. Similar to Step 1, we substitute -5 for g(x) in the function equation:

-5 = 3 - 8x

Now, we solve for x. First, subtract 3 from both sides:

-8 = -8x

Next, divide both sides by -8:

x = -8 / -8

x = 1

So, when g(x) = -5, x = 1. This step mirrors the process we employed in Step 1, but with a different output value. Again, we begin by substituting the known output, g(x) = -5, into the function equation g(x) = 3 - 8x. This transforms the problem into an algebraic equation that we can solve for x. Our goal remains the same: to isolate x on one side of the equation. The first step involves removing the constant term, 3, from the right side. We accomplish this by subtracting 3 from both sides, maintaining the balance of the equation. This gives us -8 = -8x. The next step is to eliminate the coefficient, -8, that is multiplying x. We do this by dividing both sides of the equation by -8. This isolates x and yields the solution x = 1. This means that when the input x is 1, the function g(x) produces an output of -5. We can verify this by substituting x = 1 back into the original equation: g(1) = 3 - 8(1) = 3 - 8 = -5. This confirms our solution. The ability to solve for the input x given the output g(x) is a crucial skill in understanding the inverse relationship of a function. It allows us to answer questions like, "What input do I need to get a specific output?" This skill is essential in various applications, such as finding the break-even point in economics or determining the required dosage of a medication in medicine. Just as in Step 1, the core principle here is the manipulation of algebraic equations to isolate the desired variable. Understanding the properties of equality and applying inverse operations are fundamental tools in this process. With this third value successfully determined, we move on to the final missing value, where we are given an input and asked to find the corresponding output, a process similar to Step 2.

Step 4: Finding g(x) when x = 3

In the fourth row, we are given x = 3 and need to find g(x). As in Step 2, we substitute 3 for x in the function equation:

g(3) = 3 - 8(3)

g(3) = 3 - 24

g(3) = -21

So, when x = 3, g(x) = -21. This final step reinforces the process of evaluating a function at a given input value. We are asked to find the output g(x) when the input x is 3. This involves directly substituting 3 for x in the function equation g(x) = 3 - 8x. Following the order of operations, we first perform the multiplication: 8 multiplied by 3 equals 24. This gives us g(3) = 3 - 24. Then, we perform the subtraction: 3 minus 24 equals -21. Therefore, the output g(3) is -21. This means that when the input is 3, the function produces an output of -21. This point, (3, -21), would lie on the graph of the function. Evaluating functions at specific points is a fundamental skill in mathematics and its applications. It allows us to understand the function's behavior at different inputs and to predict the outputs for various scenarios. In the context of real-world modeling, this skill is invaluable. For example, if g(x) represented the profit of a company after producing x units, then g(3) would represent the profit after producing 3 units. Similarly, if g(x) represented the distance traveled by a car after x hours, then g(3) would represent the distance traveled after 3 hours. This step, like Step 2, highlights the straightforwardness of evaluating a function when the input is known. It reinforces the importance of substitution and following the order of operations. With this final missing value determined, we have successfully completed the input-output table for the function g(x) = 3 - 8x.

The Completed Table

Now that we've calculated all the missing values, let's present the completed table:

This completed table provides a clear picture of how the function g(x) = 3 - 8x transforms different input values into their corresponding output values. The table now provides a comprehensive overview of the function's behavior for the selected input values. Each row represents a specific input-output pair, illustrating the relationship defined by the function's rule. The first row, (3/8, 0), shows the input value that results in an output of 0, also known as the root or zero of the function. The second row, (0, 3), represents the y-intercept, the point where the function's graph crosses the y-axis. The third row, (1, -5), and the fourth row, (3, -21), showcase how the function produces negative outputs for positive inputs. By examining this table, we can gain insights into the function's linearity and its decreasing nature (as the input increases, the output decreases). We can also use this table to plot these points on a graph and visualize the function's behavior. The completed table serves as a valuable tool for understanding the function's characteristics and its potential applications. It allows us to quickly look up the output for a given input or, conversely, to identify the input that produces a specific output. This is particularly useful in scenarios where we need to make predictions or analyze trends. For instance, if g(x) represented the temperature at time x, we could use this table to estimate the temperature at different times. Similarly, if g(x) represented the cost of producing x items, we could use the table to determine the cost for various production levels. The completed table is more than just a collection of numbers; it is a concise representation of the function's behavior within the specified input range. It allows us to move beyond the abstract equation and grasp the concrete relationship between inputs and outputs. This understanding is crucial for applying functions to solve real-world problems and to further explore more advanced mathematical concepts. In essence, the completed table is a visual aid that enhances our comprehension of the function g(x) = 3 - 8x and its properties.

Conclusion

Completing input-output tables for functions is a fundamental skill in mathematics. By understanding the function rule and applying basic algebraic techniques, we can effectively determine missing input or output values. This process not only solidifies our understanding of functions but also provides a foundation for more advanced mathematical concepts. In this article, we successfully completed the table for g(x) = 3 - 8x, demonstrating a step-by-step approach that can be applied to other functions as well. The ability to work with functions and input-output tables is essential for anyone pursuing mathematics, science, engineering, or any field that relies on quantitative analysis. Functions are the workhorses of mathematical modeling, and understanding their behavior is crucial for solving real-world problems. The process of completing input-output tables reinforces the fundamental concept of a function as a mapping between inputs and outputs. It highlights the importance of the function rule, which dictates how the input is transformed into the output. By working through examples like the one presented in this article, we develop a deeper appreciation for the power and versatility of functions. Furthermore, the skills acquired in completing input-output tables extend beyond the realm of pure mathematics. They are applicable in various fields, such as computer science, where functions are the building blocks of programs, and in data analysis, where functions are used to model trends and relationships. The ability to manipulate equations, solve for unknowns, and evaluate functions are all essential skills for problem-solving in a wide range of contexts. In conclusion, mastering the art of completing input-output tables is an investment in one's mathematical literacy and a valuable asset for navigating a world increasingly reliant on quantitative reasoning. The steps and techniques outlined in this article provide a solid foundation for further exploration of functions and their applications. As we continue our mathematical journey, let us remember the fundamental role of functions and the importance of understanding their behavior through various representations, including input-output tables.