Finding The Value Of F(6) For The Function F(x) = (x^2 - 4x) / 20

Hey guys! Let's dive into this math problem where we're given a function and asked to find its value at a specific point. This kind of question is super common in algebra, and it's all about understanding how functions work. So, let's break it down step by step.

The Function: f(x) = (x^2 - 4x) / 20

First off, let's take a good look at our function: f(x) = (x^2 - 4x) / 20. What this means is that for any input x, we're going to perform some operations on it and get an output. The operations here involve squaring x, subtracting four times x, and then dividing the whole thing by 20. Functions are like little machines: you feed them an input, they do their thing, and then they spit out an output. Think of it like a vending machine – you put in your money (the input), and it gives you your snack (the output).

Understanding the components of the function is crucial. The x^2 term means x multiplied by itself. For example, if x is 3, then x^2 is 3 * 3 = 9. The -4x term means -4 multiplied by x. So, if x is 3, then -4x is -4 * 3 = -12. The whole expression (x^2 - 4x) is what we get after doing these operations. Finally, we divide the result by 20. This division scales down the result, making the output a fraction or a smaller number.

Why are functions important? Functions are the backbone of mathematics and programming. They allow us to model relationships between different quantities. For example, a function could represent the height of a ball thrown in the air as a function of time, or the cost of producing items in a factory as a function of the number of items made. By understanding functions, we can make predictions and solve real-world problems. They help us see how things change and interact with each other. Imagine trying to design a bridge without understanding how the load on the bridge affects its structure – that's where functions come in! They let us model these complex relationships and ensure everything stays safe and sound. Plus, in computer science, functions are used to organize code into reusable blocks, making programs easier to write and understand.

Common mistakes when working with functions often involve misinterpreting the order of operations or messing up the signs. Always remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Also, pay close attention to negative signs, as they can easily trip you up. For instance, when squaring a negative number, remember that the result is positive (e.g., (-2)^2 = 4). Another common mistake is not substituting the value correctly into the function. Double-check your work to make sure you've replaced every instance of x with the given value. Practice makes perfect, so the more you work with functions, the more comfortable you'll become with them. Keep at it, and you'll be a function master in no time!

The Question: What is the value of f(6)?

Now, let's tackle the specific question: What is the value of f(6)? This is asking us to find the output of our function when the input is 6. In other words, we need to plug in 6 for x in the function and simplify. This is a straightforward substitution problem, but it's essential to be careful with the arithmetic.

Substituting x = 6 into the function means we replace every x in the function's formula with the number 6. So, our function f(x) = (x^2 - 4x) / 20 becomes f(6) = (6^2 - 4 * 6) / 20. This step is crucial because it sets up the rest of the calculation. If you make a mistake here, the entire answer will be incorrect. Always double-check to ensure you've correctly replaced all the x variables with the given value.

Why is substitution important in mathematics? Substitution is a fundamental technique used throughout mathematics. It allows us to evaluate expressions, solve equations, and simplify complex problems. Think of it as a way of translating a general formula into a specific result. For example, in physics, you might use substitution to calculate the speed of an object given its distance and time. In chemistry, you might use it to determine the concentration of a solution. The ability to substitute values correctly is a cornerstone of mathematical proficiency. It's like having a universal tool that can be applied in countless situations. From simple arithmetic to advanced calculus, substitution is your trusty sidekick.

Real-world examples of function evaluation are everywhere. Imagine you're calculating the cost of a taxi ride. The fare might be a function of the distance traveled, where the distance is the input, and the fare is the output. Or consider a recipe where the baking time is a function of the oven temperature. The temperature is the input, and the baking time is the output. These examples show that functions are not just abstract mathematical concepts; they are tools that help us understand and interact with the world around us. Whenever you see a relationship between two things where one depends on the other, you're likely looking at a function in action.

Solving for f(6)

Okay, let's get down to the nitty-gritty and actually calculate f(6). Remember, we have f(6) = (6^2 - 4 * 6) / 20. Now we just need to simplify this expression using the order of operations.

Step-by-step breakdown of the calculation is key to avoiding errors. First, we tackle the exponent: 6^2 = 6 * 6 = 36. Next, we perform the multiplication: 4 * 6 = 24. Now our expression looks like f(6) = (36 - 24) / 20. Subtracting inside the parentheses gives us 36 - 24 = 12. So, we have f(6) = 12 / 20. Finally, we simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4. This gives us f(6) = 3 / 5. And there you have it! We've found the value of f(6).

Different ways to simplify fractions are useful to know. Sometimes, you might not immediately see the greatest common divisor. In that case, you can simplify in stages. For example, in the fraction 12/20, you could first divide both numbers by 2, getting 6/10, and then divide by 2 again to reach 3/5. Another method is to prime factorize both the numerator and denominator and cancel out common factors. For instance, 12 = 2 * 2 * 3 and 20 = 2 * 2 * 5, so you can cancel out two 2s, leaving you with 3/5. Understanding these different techniques can make simplifying fractions less daunting.

Common mistakes in arithmetic can easily derail your calculations. It's super important to double-check your work at each step. A common mistake is messing up the signs, especially when dealing with subtraction. Another is forgetting the order of operations. Remember PEMDAS/BODMAS! Also, when simplifying fractions, make sure you're dividing both the numerator and the denominator by the same number. If you only divide one, you'll end up with the wrong answer. Practicing arithmetic regularly and paying attention to detail will help you avoid these pitfalls.

The Answer: 3/5

So, after all that, we've found that the value of f(6) is 3/5. Looking at the answer choices, we see that this matches one of the options. High five!

Checking your answer is a critical step in problem-solving. It's like having a built-in safety net. There are several ways to check your work. One way is to plug your answer back into the original equation and see if it holds true. In this case, we could substitute 3/5 back into the function and see if it gives us 6. Another approach is to work through the problem using a different method. For example, we could have simplified the function f(x) = (x^2 - 4x) / 20 before substituting x = 6. By verifying your answer, you can catch mistakes and gain confidence in your solution.

Why is understanding the process important, not just the answer? It's tempting to focus solely on getting the right answer, but understanding the process is where the real learning happens. When you understand the steps involved in solving a problem, you can apply those steps to other problems. You develop problem-solving skills that go beyond a single question. Think of it like learning to ride a bike. You don't just memorize the act of riding; you learn the principles of balance and coordination. Similarly, in math, understanding the process allows you to adapt to different situations and tackle new challenges. Plus, explaining the process to someone else reinforces your own understanding.

Different ways to approach similar problems can broaden your mathematical toolkit. For example, instead of directly substituting x = 6, we could have first simplified the function by factoring out an x from the numerator: f(x) = x(x - 4) / 20. Then, we could substitute x = 6 into the simplified function: f(6) = 6(6 - 4) / 20 = 6 * 2 / 20 = 12 / 20 = 3 / 5. Learning different approaches not only gives you more flexibility but also deepens your understanding of the underlying concepts. It's like having multiple routes to reach the same destination – you're better prepared for any obstacles along the way.

Wrapping Up

We've walked through how to evaluate a function at a specific point, step by step. Remember, functions are fundamental to math, and mastering them opens the door to all sorts of cool stuff. Keep practicing, and you'll become a function whiz in no time! This problem highlights the importance of understanding function notation, substitution, and simplification. By breaking down the problem into manageable steps and double-checking our work, we were able to arrive at the correct answer. Keep practicing these skills, and you'll be well-prepared for any similar questions that come your way. And remember, math isn't just about getting the right answer; it's about understanding the journey and the underlying concepts.