Input-Output Table: Solve G(x) = 3 - 8x

Let's dive into completing the input-output table for the function g(x) = 3 - 8x. This is a fantastic way to understand how functions work and how different inputs affect the output. We'll break down each step, making it super easy to follow along. Whether you're a student tackling homework or just brushing up on your math skills, you've come to the right place!

Understanding Input-Output Tables

Before we jump into the calculations, let's quickly recap what an input-output table is. Think of a function like a machine: you feed it an input (x), and it spits out an output (g(x)). An input-output table simply organizes these pairs, showing you what output corresponds to each input. Understanding input-output tables is fundamental in grasping the behavior of functions, which is a core concept in mathematics. These tables help visualize how a function transforms values, making it easier to identify patterns and relationships.

The function we're working with is g(x) = 3 - 8x. This means that for any value we plug in for x, we'll multiply it by -8, and then add 3 to the result. It's a linear function, which means its graph would be a straight line. The beauty of mathematics lies in its precision; each input leads to a specific, predictable output. This predictability is what allows us to model real-world phenomena using functions.

In the context of problem-solving, mastering input-output tables is essential for various applications. From calculating the trajectory of a projectile in physics to determining the cost of production in economics, functions play a crucial role. Being able to efficiently compute and interpret these tables enhances your analytical skills and equips you with a powerful tool for tackling complex problems.

Completing the Table

We have the following table to complete:

Our mission is to fill in the missing g(x) values for the inputs x = 0 and x = 3. To do this, we'll simply substitute these values into our function g(x) = 3 - 8x.

Calculating g(0)

Let's start with x = 0. We plug this value into our function:

g(0) = 3 - 8(0)

Any number multiplied by 0 is 0, so we have:

g(0) = 3 - 0

Therefore:

g(0) = 3

This tells us that when the input is 0, the output of our function is 3. This point, (0, 3), is the y-intercept of the line represented by this function. Understanding intercepts is crucial for graphing functions and visualizing their behavior. The y-intercept specifically shows where the line crosses the vertical axis on a graph, providing a key reference point.

The simplicity of this calculation highlights the elegance of functions. A single input leads to a single, well-defined output. This predictability is what makes functions so powerful in modeling real-world situations. Whether you're calculating the trajectory of a ball or predicting the growth of a population, functions provide a framework for understanding and quantifying relationships.

Calculating g(3)

Now, let's tackle x = 3. Again, we substitute this value into our function:

g(3) = 3 - 8(3)

First, we multiply 8 by 3:

g(3) = 3 - 24

Then, we subtract 24 from 3:

g(3) = -21

So, when the input is 3, the output is -21. This demonstrates how the function transforms the input value through multiplication and subtraction. The result, -21, is significantly different from the input, illustrating the function's impact.

This calculation also showcases the importance of following the order of operations (PEMDAS/BODMAS). Multiplication comes before subtraction, ensuring we arrive at the correct answer. A small mistake in the order can lead to a completely different outcome, emphasizing the need for precision in mathematical calculations.

The Completed Table

Now that we've calculated the missing values, let's fill in the table:

We've successfully completed the input-output table for the function g(x) = 3 - 8x! This completed table provides a clear picture of how the function behaves for the given inputs. Each pair of values represents a point on the line that the function describes. By plotting these points on a graph, you could visualize the linear relationship.

Notice how the g(x) values change as x increases. This pattern is a direct consequence of the function's equation. The slope of the line, which is -8 in this case, dictates how steeply the line descends. Understanding this relationship between the equation and the table (and ultimately the graph) is key to mastering functions.

This exercise underscores the value of methodical problem-solving. By breaking down the task into smaller steps – substituting values and performing arithmetic – we arrived at the solution. This approach is applicable not only in mathematics but also in many other areas of life.

Importance of Input-Output Tables

Input-output tables are more than just a way to organize numbers. They are a powerful tool for understanding functions and their behavior. They provide a visual representation of the relationship between inputs and outputs, making it easier to identify patterns and make predictions. The creation and interpretation of these tables form a cornerstone of mathematical literacy.

In various fields, input-output analysis is used to model complex systems. Economics, for instance, employs input-output models to analyze the interdependencies between different industries. Environmental science uses similar models to study ecosystems. The ability to construct and analyze these tables is therefore a valuable skill in many disciplines.

Furthermore, input-output tables lay the groundwork for more advanced mathematical concepts. Understanding how functions transform values is crucial for calculus, where we study rates of change and optimization. They also serve as a foundation for understanding computer programming, where functions are used extensively to perform specific tasks.

Conclusion

Congratulations! You've successfully completed the input-output table for the function g(x) = 3 - 8x. You've seen how to substitute input values into a function and calculate the corresponding outputs. This skill is fundamental to understanding functions and their applications in mathematics and beyond. Keep practicing, and you'll become a function whiz in no time! For further learning and exploration of mathematical functions, visit a trusted resource like Khan Academy's Algebra 1 section. Happy calculating!