Calculating G(7) For The Function G(x) = (7/8)x - 1/2 A Step-by-Step Guide

Introduction

In this article, we will explore how to determine the value of g(7) for the given function g(x) = (7/8)x - 1/2. This is a fundamental concept in algebra, involving the substitution of a specific value into a function to find the corresponding output. We'll walk through the steps in detail to ensure a clear understanding of the process. This article will guide you through the process, providing a step-by-step solution and explanation. Understanding how to evaluate functions is a crucial skill in mathematics, and this example will help solidify your grasp of the concept. We will break down the problem, making it easy to follow and understand each step. By the end of this article, you'll be confident in your ability to solve similar problems involving function evaluation. Our goal is to provide a clear and concise explanation, ensuring that you can easily apply this knowledge to future mathematical challenges. Let’s dive in and unravel the solution together!

Understanding the Function

The function provided is a linear function, which is a type of algebraic function. Specifically, we are given g(x) = (7/8)x - 1/2. This means that for any value we substitute for x, we will multiply it by 7/8 and then subtract 1/2 from the result. To find g(7), we simply replace x with 7 in the equation. Understanding the structure of the function is key to solving the problem correctly. The function g(x) is in the form y = mx + b, where m is the slope (7/8) and b is the y-intercept (-1/2). This linear form indicates that the graph of the function would be a straight line. Recognizing this form helps us understand the behavior of the function and predict its values for different inputs. In our case, we are interested in finding the value of the function when x equals 7. This involves a straightforward substitution and arithmetic calculation. The simplicity of this linear function allows us to focus on the core concept of function evaluation without getting bogged down in complex algebra.

Step-by-Step Calculation

To find g(7), we substitute x with 7 in the function g(x) = (7/8)x - 1/2. This gives us g(7) = (7/8)(7) - 1/2. Now, we perform the multiplication first: (7/8) * 7 = 49/8. Next, we subtract 1/2 from 49/8. To do this, we need a common denominator, which is 8. So, we convert 1/2 to 4/8. Thus, the equation becomes g(7) = 49/8 - 4/8. Finally, we subtract the numerators: 49 - 4 = 45. Therefore, g(7) = 45/8. This step-by-step approach ensures clarity and reduces the chance of errors. Each operation is performed sequentially, making the calculation easy to follow. The initial substitution is the crucial first step, setting the stage for the rest of the problem. The multiplication of 7/8 by 7 is a straightforward fraction multiplication, and the subsequent subtraction requires finding a common denominator, a fundamental skill in arithmetic. By breaking down the calculation into smaller, manageable steps, we can confidently arrive at the correct answer. This method is particularly helpful for learners who are new to function evaluation, as it emphasizes the importance of careful and organized calculation.

Detailed Solution

The process of finding g(7) involves several key steps, each crucial to arriving at the correct answer. Initially, we have the function g(x) = (7/8)x - 1/2. Our task is to determine the output of this function when the input x is 7. This means we need to substitute 7 for every instance of x in the function’s equation. Substituting x = 7 into the function, we get g(7) = (7/8)(7) - 1/2. This substitution is the foundational step, converting the general function notation into a specific calculation. The next step involves performing the multiplication. We multiply the fraction 7/8 by the whole number 7. This is done by multiplying the numerator (7) by 7, resulting in 49, while the denominator (8) remains the same. Thus, (7/8)(7) equals 49/8. Our equation now looks like this: g(7) = 49/8 - 1/2. The subsequent step is to subtract 1/2 from 49/8. Before we can subtract these fractions, we need a common denominator. The least common denominator for 8 and 2 is 8. So, we need to convert 1/2 into an equivalent fraction with a denominator of 8. To do this, we multiply both the numerator and the denominator of 1/2 by 4. This gives us 1/2 = (1 * 4) / (2 * 4) = 4/8. Now our equation is g(7) = 49/8 - 4/8. With a common denominator, we can now subtract the fractions. We subtract the numerators (49 - 4) while keeping the denominator the same. This gives us 45/8. Therefore, the value of g(7) is 45/8. This detailed walkthrough highlights each step, ensuring clarity and understanding. From the initial substitution to the final subtraction, every operation is explained, making the process accessible to learners of all levels. The meticulous approach reduces the likelihood of errors and reinforces the fundamental principles of function evaluation.

Evaluating the Answer Choices

Now that we've calculated g(7) = 45/8, let's compare this result to the provided answer choices. The options are:

A. 60/7 B. 45/8 C. 49/8 D. 53/8

By comparing our calculated value of 45/8 with the answer choices, we can clearly see that option B, 45/8, matches our result. Therefore, the correct answer is B. This step is crucial to ensure that our calculated answer aligns with the provided options, reinforcing the accuracy of our solution. In a multiple-choice scenario, verifying your result against the available choices is a best practice that can help catch any potential errors in your calculations. By systematically comparing our answer with each option, we can confidently select the correct choice. This process also provides an opportunity to double-check our work, ensuring that we have not made any mistakes along the way. In this case, the match between our calculated value and option B confirms the validity of our solution, giving us assurance that we have correctly evaluated the function at x = 7.

Conclusion

In summary, to find the value of g(7) for the function g(x) = (7/8)x - 1/2, we substituted x with 7, performed the necessary arithmetic operations, and arrived at the result g(7) = 45/8. This exercise demonstrates the fundamental process of evaluating a function at a specific point. Understanding how to substitute values into functions and simplify the resulting expressions is a core skill in algebra and calculus. This example provides a clear, step-by-step illustration of this process, making it accessible to students and learners of all levels. The key to success in these types of problems lies in careful substitution, accurate arithmetic, and a systematic approach. By following the steps outlined in this article, you can confidently tackle similar problems and deepen your understanding of function evaluation. Mastering this skill is essential for further studies in mathematics and its applications in various fields. The ability to evaluate functions is a building block for more advanced mathematical concepts, and this article aims to provide a solid foundation for your future learning.