Evaluating F(-142) For F(x) = -0.2(x + 223) A Step-by-Step Solution

by ADMIN 68 views

Hey guys! Let's dive into a math problem where we need to find the value of a function at a specific point. This is a common type of question in algebra, and it's all about plugging in values and doing some simple arithmetic. So, let's break it down step by step.

Understanding the Function

First, let's understand the function we're working with: f(x) = -0.2(x + 223). This is a linear function, which means it represents a straight line when graphed. The function takes an input x, adds 223 to it, and then multiplies the result by -0.2. Our goal is to find the output of this function when x is -142. In other words, we want to find f(-142).

This function is presented in slope-intercept form, although it might not look like it at first glance. We can rewrite it to the more familiar y = mx + b form if we distribute the -0.2. However, for our purpose of evaluating the function at a specific point, it's perfectly fine as it is. The key here is to recognize that f(x) is simply a rule that tells us what to do with the input x. Understanding the structure of the function helps us approach the problem with confidence.

The coefficient -0.2 plays a crucial role in determining the function's behavior. It indicates the slope of the line represented by the function. Since it's negative, we know the line will be decreasing as x increases. This is a useful piece of information that can help us anticipate the sign of our final answer. Also, the constant 223 inside the parentheses shifts the graph horizontally. These details, while not directly used in the calculation, provide a deeper understanding of the function's properties.

Before we jump into plugging in the value, it's worth thinking about what kind of answer we might expect. Since we're adding 223 to -142, we'll get a positive number inside the parentheses. Multiplying that positive number by -0.2 will give us a negative result. This simple prediction can help us catch any mistakes we might make during the calculation. It's always a good idea to have a rough estimate in mind before you start crunching the numbers.

Step-by-Step Evaluation

Now, let's evaluate the function step by step. Here's how we do it:

  1. Substitute x with -142 in the function: f(-142) = -0.2(-142 + 223)
  2. Simplify the expression inside the parentheses: -142 + 223 = 81 So, we now have: f(-142) = -0.2(81)
  3. Multiply -0.2 by 81: -0.2 * 81 = -16.2

Therefore, f(-142) = -16.2.

Each step in the evaluation is crucial. The substitution replaces the variable with the specific value we're interested in. The simplification inside the parentheses follows the order of operations (PEMDAS/BODMAS), ensuring we handle the addition before the multiplication. Finally, the multiplication gives us the function's output at the given input. Paying attention to the signs is especially important when dealing with negative numbers, as a small mistake can lead to a wrong answer.

The arithmetic involved might seem simple, but accuracy is key. Double-checking each step can help prevent errors. For instance, we can quickly verify that -142 + 223 indeed equals 81. Similarly, we can mentally estimate -0.2 * 81 as roughly -0.2 * 80, which is -16, confirming that our result of -16.2 is reasonable. These small checks can save us from selecting the wrong answer.

Another way to think about this process is to imagine a machine. The function f(x) is like a machine that takes a number as input, performs some operations on it, and spits out a new number. In our case, we're feeding the machine the number -142, and it's churning out -16.2. This analogy can be helpful in visualizing how functions work and what it means to evaluate them.

The Correct Answer

Looking at the options, we see that the correct answer is:

B. 16.2

Wait a minute! Did we make a mistake? Our calculation resulted in -16.2, not 16.2. This highlights the importance of paying close attention to the signs. It's easy to overlook a negative sign, especially under pressure. Let's re-examine our steps to make sure we didn't make an error.

Ah, we found it! The correct answer should be -16.2, but that's not one of the options provided. This indicates there might be an error in the question itself or the answer choices. In a real-world scenario, this is a valuable lesson: don't blindly choose an answer just because it's there. If your calculations are correct and none of the options match, it's possible there's a mistake in the question.

In this case, if we were taking a test, we would likely bring this discrepancy to the attention of the instructor. It's crucial to be confident in your work and not second-guess yourself when you've followed the correct procedure. Math is about precision, and sometimes, the correct answer isn't explicitly listed.

If we had to choose the closest answer, we might be tempted to pick B. 16.2, but it's essential to recognize that it's not the correct answer. This situation underscores the importance of understanding the concepts and the calculations, rather than simply trying to match an answer choice. Always trust your process and your results, and don't be afraid to question the given options if they don't align with your findings.

Key Takeaways

Alright, guys, key takeaways from this problem:

  • Substituting values into functions is a fundamental skill in algebra.
  • Pay close attention to signs, especially when dealing with negative numbers.
  • Follow the order of operations (PEMDAS/BODMAS) to ensure accurate calculations.
  • Double-check your work to catch any errors.
  • Don't be afraid to question the answer choices if they don't match your results.

This type of problem appears frequently in algebra and precalculus courses. Mastering the skill of function evaluation is crucial for more advanced topics like calculus and beyond. The process involves not just plugging in numbers, but also understanding the function's behavior and interpreting the results in context.

Moreover, the ability to identify potential errors in the problem itself is a valuable skill. It demonstrates a deep understanding of the material and a critical thinking approach. In mathematics, it's not enough to simply find an answer; it's equally important to be able to justify your solution and recognize when something doesn't seem quite right.

Practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! Remember, math is like a puzzle, and each piece needs to fit perfectly to get the complete picture.

Conclusion

In conclusion, evaluating functions is a core concept in mathematics. By carefully substituting the given value, simplifying the expression, and paying attention to detail, we can arrive at the correct answer. And remember, it's okay to question the provided options if they don't align with your accurate calculations. Keep practicing, and you'll become more confident in your math skills! You've got this!

Let's tackle the problem of correctly evaluating the function f(x) = -0.2(x + 223) at x = -142. This is a fundamental skill in algebra, and we'll walk through the process step-by-step to ensure clarity and accuracy. Guys, understanding how to evaluate functions is essential for various mathematical concepts, so let’s get this right!

Understanding the Function and the Task

First, let’s make sure we understand the function f(x) = -0.2(x + 223). This function tells us to take an input x, add 223 to it, and then multiply the result by -0.2. Our task is to find the value of f(-142), which means we need to substitute -142 for x in the function and simplify the expression.

The function itself is a linear function, which means it represents a straight line when graphed. The -0.2 coefficient indicates the slope of the line, and the +223 within the parentheses is related to the horizontal shift of the graph. While we don't need to graph the function to evaluate it at x = -142, understanding these properties can provide valuable context and help us visualize the function's behavior. The key is to see the function as a rule that transforms an input value into an output value. In our case, we're feeding in -142 and want to see what the function spits out.

Before we dive into the calculations, it's a good idea to make a rough estimate of the result. This can help us catch any major errors along the way. We're adding 223 to -142, which should give us a positive number. Then we're multiplying by -0.2, which means the final result should be negative. Keeping this in mind will help us check if our answer is reasonable.

Another way to think about this is to break down the function into smaller steps. First, we perform the addition inside the parentheses. Then, we multiply the result by -0.2. This stepwise approach can make the process less daunting and reduce the chances of making a mistake. Each step is a small, manageable calculation that contributes to the final answer. It's like building a house brick by brick, ensuring each one is placed correctly before moving on to the next.

Step-by-Step Evaluation of f(-142)

Now, let’s evaluate f(-142) step by step. This is where we put our understanding into action.

  1. Substitute x with -142: f(-142) = -0.2(-142 + 223)
    • This is the crucial first step where we replace the variable x with the specific value we're interested in. It's like plugging the number -142 into the function machine. Be careful to keep track of the signs – a common source of errors.
  2. Simplify the expression inside the parentheses: -142 + 223 = 81
    • Here, we perform the addition inside the parentheses. Remember to follow the order of operations (PEMDAS/BODMAS). The result, 81, is a positive number, which aligns with our earlier estimate.
  3. Multiply -0.2 by 81: -0.2 * 81 = -16.2
    • Finally, we multiply the result from the previous step by -0.2. This gives us -16.2, which is a negative number, as we expected.

Therefore, f(-142) = -16.2. We've successfully evaluated the function at the given point.

Each step is like a checkpoint, ensuring we're on the right track. We can quickly verify each calculation: -142 + 223 does indeed equal 81, and -0.2 multiplied by 81 gives us -16.2. This methodical approach minimizes the risk of errors and builds confidence in our answer. It's like proofreading a document before submitting it – a small investment of time that can prevent costly mistakes.

Checking the Answer Choices

Now, let’s check the answer choices. We have:

  • A. 81
  • B. 16.2
  • C. 251.4
  • D. -1866

Notice that none of the options is -16.2, which is what we calculated. This means either we made a mistake, or there's an error in the question or the answer choices. It's crucial not to panic at this point. Instead, let's calmly review our work to see if we can identify any errors.

This situation highlights the importance of being a critical thinker in mathematics. It's not enough to simply go through the motions; we need to be able to evaluate our results and identify potential discrepancies. If our calculations are correct and none of the options match, it's a sign that something might be amiss with the question itself. In a real-world test scenario, this would be the time to raise your hand and ask the instructor for clarification.

It's tempting to choose the closest answer, which in this case would be B. 16.2. However, it's essential to recognize that this is not the correct answer. Choosing B would be settling for an approximation, rather than sticking with the precise result we obtained. This distinction is crucial in mathematics, where accuracy is paramount.

What to Do When No Answer Matches

So, what should we do when no answer matches our correct calculation? Here are a few steps we can take:

  1. Review our steps: Carefully go back through each step of our calculation to ensure we haven't made any arithmetic errors or sign mistakes.
  2. Re-evaluate the problem: Make sure we understand the function and the task correctly. Are we substituting the value correctly? Are we following the order of operations?
  3. Consider the possibility of an error in the question: If we've checked our work thoroughly and we're confident in our answer, it's possible that there's an error in the question itself or the answer choices.
  4. Seek clarification: If possible, ask the instructor or test proctor for clarification.

In this case, we've already reviewed our steps and we're confident that our calculation is correct. Therefore, the most likely scenario is that there's an error in the answer choices. In a test situation, we would bring this to the attention of the instructor.

This situation also provides a valuable lesson: Don't blindly trust the answer choices. Always rely on your own calculations and reasoning. If you've done the work correctly, you should be confident in your answer, even if it doesn't match the options provided.

Key Skills and Takeaways

Alright guys, let’s talk about key skills and takeaways from this exercise:

  • Function evaluation: We've practiced the fundamental skill of evaluating a function at a specific point. This is a crucial skill in algebra and calculus.
  • Order of operations: We've reinforced the importance of following the order of operations (PEMDAS/BODMAS) to ensure accurate calculations.
  • Attention to detail: We've seen how easily a small sign mistake can lead to an incorrect answer. Paying attention to detail is essential in mathematics.
  • Critical thinking: We've learned how to evaluate our results and identify potential discrepancies. This is a valuable skill that goes beyond just getting the right answer.
  • Problem-solving: We've developed a strategy for what to do when no answer choices match our calculation. This involves reviewing our work, re-evaluating the problem, and considering the possibility of an error in the question.

These skills are not just important for math class; they're transferable to many other areas of life. Being able to think critically, pay attention to detail, and solve problems effectively are valuable assets in any field.

Conclusion and Final Thoughts

In conclusion, evaluating f(-142) for the function f(x) = -0.2(x + 223) gives us -16.2. However, since this answer is not among the provided options, it's likely that there's an error in the question itself. This exercise has highlighted the importance of not only performing calculations accurately but also thinking critically about the results and being prepared to question the given information. Keep practicing these skills, and you'll become a more confident and effective problem-solver. You've got this!

Hey everyone! Let's walk through a step-by-step solution for evaluating the function f(x) = -0.2(x + 223) at x = -142. This type of problem is a cornerstone of algebra, and mastering it will set you up for success in more advanced math courses. So, grab your pencils, and let's get started! The most important skill here is paying attention to each step to prevent small errors.

Understanding Function Evaluation

First, let's understand what it means to evaluate a function. A function is like a machine that takes an input, does something to it, and gives you an output. In this case, our function f(x) = -0.2(x + 223) takes an input x, adds 223 to it, and then multiplies the result by -0.2. When we're asked to find f(-142), we're essentially being asked: "What output does this function give us when we input -142?"

Thinking of a function as a machine can be really helpful. Imagine a vending machine – you put in money (the input), press a button (the function's rule), and out comes your snack (the output). Similarly, with our function, we're putting in -142, and we need to figure out what comes out. This concept is fundamental to understanding functions and their applications in various fields.

The expression f(x) is simply a notation for the output of the function. It's the "y" in the familiar y = mx + b form of a linear equation. In our case, we can think of f(x) as the "y-value" corresponding to the "x-value" -142. Understanding this notation helps us communicate mathematical ideas clearly and concisely.

Before we start plugging in numbers, it's always a good idea to get a sense of what the answer might look like. We're adding 223 to -142, which will give us a positive number. Then we're multiplying by -0.2, which will make the final result negative. This quick mental check helps us catch any major errors later on.

Another way to visualize this is to consider the graph of the function. Since it's a linear function, it will be a straight line. The negative coefficient (-0.2) tells us that the line will be sloping downwards. Knowing this can help us interpret the result we get in the context of the function's overall behavior.

Step-by-Step Solution

Now, let's dive into the step-by-step solution:

  1. Substitute x with -142: f(-142) = -0.2(-142 + 223)
    • This is the most crucial first step. We're replacing the variable x with the specific value we want to evaluate the function at. Think of it as plugging -142 into the function machine. Be extra careful with the signs at this stage – a misplaced negative can throw off the entire calculation.
  2. Simplify the expression inside the parentheses: -142 + 223 = 81
    • Here, we're applying the order of operations (PEMDAS/BODMAS). Parentheses come first, so we perform the addition inside the parentheses. This step involves adding a negative number to a positive number, which can be a tricky spot for some folks. Double-check your work to make sure you get the correct result.
  3. Multiply -0.2 by 81: -0.2 * 81 = -16.2
    • Finally, we multiply the result from the parentheses by -0.2. This is where we need to remember the rules of multiplying positive and negative numbers. A negative times a positive is a negative. So, our final answer will be negative. The actual multiplication can be done by hand or with a calculator. Double-checking your work is always a good idea.

Therefore, f(-142) = -16.2. We've successfully evaluated the function at the given input value.

Each step in this process is like a small victory. We've carefully substituted, simplified, and multiplied, ensuring we're on the right track. By breaking the problem down into smaller, manageable steps, we reduce the chances of making a mistake and build confidence in our solution. It's like climbing a staircase one step at a time – each step brings us closer to the top.

Analyzing the Answer Choices

Now, let's take a look at the answer choices provided:

  • A. 81
  • B. 16.2
  • C. 251.4
  • D. -1866

Notice that none of the options is -16.2, which is the result we calculated. This is a bit of a puzzle! It means either we made a mistake somewhere, or there's an error in the question itself or the provided answer choices. This is where our critical thinking skills come into play. We can't just blindly pick an answer; we need to figure out what's going on.

This situation highlights the importance of not just memorizing procedures but also understanding the underlying concepts. If we simply memorized the steps without understanding why we're doing them, we might be tempted to pick the closest answer, even if it's not correct. But because we understand the process, we can confidently say that none of the options matches our correct calculation.

So, what do we do in this situation? The first step is to carefully review our work. Did we make any arithmetic errors? Did we substitute the value correctly? Did we follow the order of operations? It's always a good idea to have a fresh pair of eyes look over your work, if possible. A friend or classmate might spot a mistake that you missed.

If we've thoroughly checked our work and we're still confident in our answer, then the most likely explanation is that there's an error in the question or the answer choices. In a real-world test scenario, this is the time to bring it to the attention of the instructor or test proctor. It's important to be respectful and clear when explaining the situation. Simply saying "The answer is not there" isn't as effective as saying "I calculated the answer to be -16.2, but that's not one of the options provided. Could there be an error in the answer choices?"

What to Do When Answers Don't Match

So, what should you do when the answers don't match your calculations? Here's a quick checklist:

  1. Double-check your work: Go through each step carefully, looking for any arithmetic errors or sign mistakes.
  2. Review the problem statement: Make sure you understand the question correctly and haven't missed any important information.
  3. Consider alternative approaches: Is there another way to solve the problem? Sometimes a different method can help you spot errors.
  4. If possible, seek clarification: Ask your instructor or a classmate for help. A fresh perspective can be invaluable.
  5. If you're confident in your answer and none of the options match, raise the issue with the instructor: It's possible there's an error in the question or the answer choices.

Remember, it's better to be confident in your correct answer, even if it's not listed, than to blindly choose an incorrect option. Math is about precision and accuracy, and it's important to trust your own reasoning and calculations.

Key Skills Reinforced

Let's highlight the key skills we've reinforced in this exercise:

  • Function evaluation: We've practiced the core skill of evaluating a function at a given input value. This is a fundamental building block for more advanced math topics.
  • Order of operations: We've emphasized the importance of following the order of operations (PEMDAS/BODMAS) to ensure accurate calculations.
  • Attention to detail: We've seen how easily small errors can creep in, and the importance of double-checking your work.
  • Critical thinking: We've developed our ability to analyze our results and identify potential discrepancies.
  • Problem-solving: We've learned a strategy for what to do when the answer choices don't match our calculations.

These skills are not just useful in math class; they're transferable to many other areas of life. Being able to think critically, pay attention to detail, and solve problems effectively are valuable assets in any field.

Final Thoughts and Conclusion

In conclusion, evaluating f(-142) for the function f(x) = -0.2(x + 223) gives us -16.2. Since this answer is not among the provided options, it's likely that there's an error in the question itself. This exercise has been a great reminder of the importance of not only performing calculations accurately but also thinking critically about our results and being prepared to question the given information. Keep practicing these skills, and you'll become a more confident and effective math student. You've got this!