Complete The Table For G(x) = 3 - 8x Input-Output Function
In mathematics, functions play a crucial role in describing relationships between variables. Understanding how to evaluate a function for different inputs is a fundamental skill. This article will guide you through the process of completing a table of inputs and outputs for the given function, g(x) = 3 - 8x. We'll explore how to determine the output (g(x)) for a given input (x) and, conversely, how to find the input (x) that produces a specific output (g(x)). This exercise will not only solidify your understanding of function evaluation but also enhance your problem-solving abilities in algebra and beyond.
Understanding Functions and Input-Output Tables
Before we dive into the specifics of the function g(x) = 3 - 8x, let's first establish a clear understanding of what functions are and how input-output tables work. At its core, a function is a mathematical rule that assigns a unique output value for each input value. Think of it like a machine: you put something in (the input), and the machine processes it according to its rule and produces something else (the output). In mathematical notation, we often represent a function as f(x), where 'f' is the name of the function and 'x' is the input variable. The output is the value of the function, often denoted as f(x). In our case, the function is g(x) = 3 - 8x. This means that for any input value 'x', the function will multiply 'x' by -8, then add 3 to the result to get the output value g(x).
An input-output table, also known as a function table, is a way to organize and visualize the relationship between inputs and outputs of a function. It typically has two columns: one for the input values (x) and one for the corresponding output values (g(x) in our case). By filling in this table, we can see how the function transforms different inputs into their respective outputs, revealing the function's behavior across various values. Input-output tables are particularly useful for understanding the function's pattern, identifying key values, and even graphing the function. They provide a structured way to explore the relationship defined by the function and are a cornerstone of mathematical analysis.
The table we're going to complete will have specific input values for 'x', and we'll calculate the corresponding output values for g(x). Conversely, we'll also be given some output values for g(x) and need to determine the input values 'x' that produce them. This bidirectional approach will provide a comprehensive understanding of the function g(x) = 3 - 8x.
Completing the Table: Finding g(x) for Given x
Let's start by tackling the first part of the problem: finding the output values, g(x), for the given input values of x. The function we're working with is g(x) = 3 - 8x. This means that to find g(x) for a particular value of x, we simply substitute that value into the expression and simplify. We'll be using the order of operations (PEMDAS/BODMAS) to ensure we arrive at the correct answer. Remember, this involves performing multiplication before addition and subtraction. The key is to carefully substitute the given 'x' value into the equation and then follow the order of operations to compute the result. This is a fundamental process in function evaluation and is crucial for understanding how functions work.
The table provides us with two instances where we need to find g(x) for a given x: when x = 0 and when x = 3. Let's work through these one at a time:
Case 1: x = 0
To find g(0), we substitute 0 for x in the function g(x) = 3 - 8x:
g(0) = 3 - 8(0)
First, we perform the multiplication: 8(0) = 0
g(0) = 3 - 0
Then, we perform the subtraction: 3 - 0 = 3
Therefore, g(0) = 3. This means that when the input is 0, the output of the function g(x) is 3.
Case 2: x = 3
Similarly, to find g(3), we substitute 3 for x in the function g(x) = 3 - 8x:
g(3) = 3 - 8(3)
First, we perform the multiplication: 8(3) = 24
g(3) = 3 - 24
Then, we perform the subtraction: 3 - 24 = -21
Therefore, g(3) = -21. This tells us that when the input is 3, the output of the function g(x) is -21.
By carefully substituting the given values of 'x' into the function and following the order of operations, we've successfully calculated the corresponding output values for g(x). These calculations demonstrate the core principle of function evaluation and are essential for completing the table.
Completing the Table: Finding x for Given g(x)
Now, let's move on to the second part of the problem: finding the input values, x, for the given output values of g(x). This is the reverse process of what we did before. Instead of substituting 'x' into the equation and solving for g(x), we'll be substituting g(x) into the equation and solving for 'x'. This involves using algebraic manipulation to isolate 'x' on one side of the equation. The key here is to remember the order of operations in reverse: we'll undo addition and subtraction first, then undo multiplication and division. This requires a solid understanding of algebraic principles and the ability to solve linear equations.
The table provides us with two instances where we need to find x for a given g(x): when g(x) = 0 and when g(x) = -5. Let's tackle these cases systematically.
Case 1: g(x) = 0
We are given that g(x) = 0. We need to find the value of x that makes this true. We start by setting the function g(x) equal to 0:
3 - 8x = 0
Our goal is to isolate 'x'. First, we subtract 3 from both sides of the equation to undo the addition of 3:
3 - 8x - 3 = 0 - 3
-8x = -3
Next, we divide both sides by -8 to undo the multiplication by -8:
-8x / -8 = -3 / -8
x = 3/8
Therefore, when g(x) = 0, x = 3/8. This means that the input value of 3/8 will produce an output value of 0 for the function g(x).
Case 2: g(x) = -5
Similarly, we are given that g(x) = -5. We need to find the value of x that satisfies this condition. We begin by setting the function g(x) equal to -5:
3 - 8x = -5
Again, our aim is to isolate 'x'. First, we subtract 3 from both sides of the equation:
3 - 8x - 3 = -5 - 3
-8x = -8
Next, we divide both sides by -8:
-8x / -8 = -8 / -8
x = 1
Therefore, when g(x) = -5, x = 1. This indicates that an input value of 1 will result in an output value of -5 for the function g(x).
By setting the function equal to the given output values and solving the resulting equations, we've successfully determined the corresponding input values. This process highlights the inverse relationship between input and output in a function and reinforces the importance of algebraic skills in mathematical problem-solving.
The Completed Table
Now that we've calculated all the missing input and output values, we can complete the table. Here's the filled-in table for the function g(x) = 3 - 8x:
| x | g(x) |
|---|---|
| 3/8 | 0 |
| 0 | 3 |
| 1 | -5 |
| 3 | -21 |
This completed table provides a clear representation of the relationship between the input values (x) and the output values (g(x)) for the function g(x) = 3 - 8x. We can see how different input values are transformed by the function, and we can also identify the input values that produce specific output values. This table serves as a valuable tool for understanding the function's behavior and can be used for various purposes, such as graphing the function or making predictions about its output for other input values.
Conclusion
Completing the table of inputs and outputs for the function g(x) = 3 - 8x has provided us with a solid understanding of function evaluation and the relationship between inputs and outputs. We've learned how to find g(x) for a given x by substituting the value into the function and simplifying. We've also learned how to find x for a given g(x) by setting up an equation and solving for x. These skills are essential for working with functions in mathematics and have broad applications in various fields.
The process of filling in the table has reinforced the importance of following the order of operations, applying algebraic principles, and carefully substituting values into equations. Furthermore, it has demonstrated the bidirectional nature of functions: we can move from input to output and from output to input. This understanding is crucial for a deeper comprehension of mathematical relationships and problem-solving.
By mastering these fundamental concepts, you'll be well-equipped to tackle more complex mathematical problems and applications involving functions. Remember to practice these skills regularly to solidify your understanding and build confidence in your abilities. Functions are a cornerstone of mathematics, and a strong grasp of their properties and behavior will serve you well in your mathematical journey.
