Evaluating (r/s)(8) With Given Functions R(x) And S(x)

Introduction

In this comprehensive guide, we will walk through the step-by-step process of evaluating the expression (r/s)(8) given the functions r(x) = 3x - 1 and s(x) = 2x + 1. This type of problem is common in algebra and precalculus, and understanding how to solve it is crucial for mastering function operations. We will break down each step, providing clear explanations and examples along the way. By the end of this article, you'll have a solid grasp of how to evaluate composite functions and handle function division effectively. So, let’s dive in and unravel this mathematical puzzle!

Understanding Function Operations

Before we jump into the problem, it's essential to understand the basics of function operations. Functions are mathematical entities that take an input, perform a specific operation, and produce an output. When we talk about operations on functions, we're referring to how we can combine functions using arithmetic operations like addition, subtraction, multiplication, and division.

Basic Arithmetic Operations on Functions

• Addition (r + s)(x): This means you add the two functions together. So, (r + s)(x) = r(x) + s(x).
• Subtraction (r - s)(x): This involves subtracting one function from another. Thus, (r - s)(x) = r(x) - s(x).
• Multiplication (r * s)(x): Here, you multiply the two functions. (r * s)(x) = r(x) * s(x).
• Division (r/s)(x): This is where you divide one function by another. Hence, (r/s)(x) = r(x) / s(x), but it's crucial to remember that s(x) cannot be zero.

In our case, we are dealing with the division of two functions, r(x) and s(x). The notation (r/s)(x) represents the function r(x) divided by the function s(x). When we evaluate (r/s)(8), we first find the expressions for r(8) and s(8) separately, and then we divide r(8) by s(8). This approach ensures we follow the correct order of operations and arrive at the accurate result.

Defining the Functions r(x) and s(x)

In our problem, we are given two functions:

• r(x) = 3x - 1
• s(x) = 2x + 1

These functions are linear functions, which means they represent straight lines when graphed. The function r(x) takes an input x, multiplies it by 3, and then subtracts 1. The function s(x) takes an input x, multiplies it by 2, and then adds 1. Understanding the specific operations each function performs is the first step in evaluating (r/s)(8).

When we need to evaluate a function at a specific point, such as r(8) or s(8), we substitute the value of x (in this case, 8) into the function's expression. This process is straightforward but crucial for correctly solving the problem. The next step involves plugging in the value 8 into both r(x) and s(x) to find the respective values.

Step-by-Step Evaluation of (r/s)(8)

Now, let's dive into the step-by-step evaluation of (r/s)(8). This process involves several key steps:

Step 1: Evaluate r(8)

To find r(8), we substitute x with 8 in the function r(x) = 3x - 1:

r(8) = 3(8) - 1 r(8) = 24 - 1 r(8) = 23

So, the value of r(8) is 23. This means that when the input to the function r(x) is 8, the output is 23. This step is fundamental as it gives us the numerator of the fraction we will eventually compute.

Step 2: Evaluate s(8)

Next, we need to find s(8). We substitute x with 8 in the function s(x) = 2x + 1:

s(8) = 2(8) + 1 s(8) = 16 + 1 s(8) = 17

Thus, s(8) equals 17. This means that when the input to the function s(x) is 8, the output is 17. This value will serve as the denominator in our final calculation.

Step 3: Calculate (r/s)(8)

Now that we have r(8) and s(8), we can find (r/s)(8). Recall that (r/s)(8) means r(8) divided by s(8):

(r/s)(8) = r(8) / s(8) (r/s)(8) = 23 / 17

Therefore, (r/s)(8) is 23/17. This is the final value of the expression. It represents the result of dividing the output of r(8) by the output of s(8). The fraction 23/17 is already in its simplest form, as 23 and 17 are both prime numbers and do not have any common factors other than 1.

Analyzing the Answer Choices

Now that we've calculated (r/s)(8) to be 23/17, let's analyze the given answer choices to see which one matches our result. The initial question provided several options, and we need to identify the correct one.

The answer choices typically present different expressions or values, and it's crucial to compare our calculated result with each option carefully. Sometimes, the answer choices might present the expression in an unsimplified form, so understanding the steps we took to arrive at our solution is vital. For instance, the correct answer choice might show the initial substitution without performing the final calculation.

Looking back at the choices:

• A. $\frac{3(6)-1}{2(6)+1}$
• B. $\frac{(\text { B })}{2(\text { b })+1}$
• C. $\frac{36-1}{26+1}$
• D. $\frac{(\text { 6) }-1}{(\text { 6 })+1}$

None of these options directly present 23/17. However, we need to consider if any of these choices are equivalent to the steps we took to arrive at our answer. Let's re-examine our calculations:

We found r(8) = 3(8) - 1 = 23 and s(8) = 2(8) + 1 = 17. Thus, (r/s)(8) = 23/17.

Comparing this with the given options, none of them directly reflect the correct substitution and calculation. There seems to be a typo or error in the provided choices, as they do not accurately represent the correct evaluation of (r/s)(8) for the given functions r(x) and s(x).

Common Mistakes and How to Avoid Them

When evaluating function operations, it's easy to make mistakes if you're not careful. Here are some common pitfalls and how to avoid them:

1. Incorrect Substitution: One of the most frequent errors is substituting the value of x incorrectly into the functions. For example, mixing up the substitution in r(x) and s(x) or miscalculating the arithmetic after substitution. To avoid this, double-check each substitution and perform the calculations step by step.
2. Order of Operations: Forgetting the correct order of operations (PEMDAS/BODMAS) can lead to incorrect results. Ensure you perform multiplication before addition or subtraction. Write out each step to keep track of the order.
3. Division by Zero: In the case of function division, always check if the denominator (s(x) in our case) could be zero. If it is, then the expression is undefined at that point. In this specific problem, s(8) = 17, so we don't have this issue, but it's a crucial check for other problems.
4. Misinterpreting Notation: Sometimes, students misinterpret the notation of function operations. For instance, they might confuse (r/s)(x) with r(x) / s(x). Make sure you understand the notation and what it represents.

By being mindful of these common mistakes and taking a methodical approach, you can improve your accuracy and confidence in solving function operation problems.

Conclusion

In this article, we've thoroughly explored how to evaluate (r/s)(8) given the functions r(x) = 3x - 1 and s(x) = 2x + 1. We started by understanding the basic arithmetic operations on functions, then defined our functions, and proceeded with a step-by-step evaluation. We found that r(8) = 23 and s(8) = 17, leading to (r/s)(8) = 23/17. While analyzing the answer choices, we noted that none of the provided options accurately matched our result, indicating a potential error in the original question.

We also discussed common mistakes to avoid, such as incorrect substitution, order of operations errors, division by zero, and misinterpreting notation. By being aware of these pitfalls and practicing a systematic approach, you can enhance your problem-solving skills in function operations.

Understanding function operations is fundamental in mathematics, especially in algebra and calculus. Mastering these concepts will not only help you in academic settings but also in various real-world applications where functions are used to model different phenomena.

For further learning and practice on function operations, consider visiting Khan Academy's Functions section. This resource provides numerous lessons, exercises, and quizzes to solidify your understanding.