Solving F(6) For F(x) = 7x - 4

Hey guys! Today we're diving into a super common math problem that pops up all the time: evaluating a function at a specific value. We're going to tackle the function $f(x) = 7x - 4$ and figure out what $f(6)$ is. This might seem straightforward, but understanding the process is key to mastering more complex functions down the line. So, let's break it down and make sure you guys feel confident about this. We'll go through each step, explain the reasoning, and even touch upon why this skill is so important in the grand scheme of mathematics and beyond. Get ready to flex those brain muscles!

Understanding Function Notation

Alright, let's kick things off by getting a solid grip on what function notation actually means. When you see something like $f(x) = 7x - 4$, it's basically a way of saying that 'f' is a rule that takes an input, which we represent with 'x', and performs a specific operation on it to give us an output. Think of it like a machine: you put something in (the input), the machine does its thing, and something comes out (the output). In our case, the 'f' machine takes 'x', multiplies it by 7, and then subtracts 4. The notation $f(x)$ tells us the name of the function is 'f', and 'x' is the variable or the input placeholder. It's a really concise way to describe relationships between numbers. It's super useful because it allows us to define complex relationships in a simple, standardized format. So, when we're asked to find $f(6)$, we're not being asked to do anything mysterious; we're simply being told to use the function 'f' and to plug in the value 6 wherever we see 'x'. This is the fundamental concept behind working with functions, and once you get this, a whole world of mathematical possibilities opens up.

The Process of Evaluating a Function

Now, let's get to the nitty-gritty of how we actually evaluate a function. Evaluating a function means finding the output value for a given input value. For our function $f(x) = 7x - 4$, we want to find $f(6)$. This means we need to substitute the number 6 for every occurrence of the variable 'x' in the function's rule. It's like saying, "Okay, 'x', you're out, and '6' is in!" So, we take our equation $f(x) = 7x - 4$ and rewrite it by replacing every 'x' with '6'. This gives us $f(6) = 7(6) - 4$. Notice that we put the 6 in parentheses to make it clear that it's replacing 'x' and that it's being multiplied by 7. Order of operations (PEMDAS/BODMAS) is crucial here. First, we perform the multiplication: $7 \times 6 = 42$. After that, we perform the subtraction: $42 - 4 = 38$. So, $f(6) = 38$. It's a step-by-step process: identify the input, substitute it into the function, and then simplify using the correct order of operations. This method applies to any function, no matter how complicated it looks. The core idea remains the same: replace the variable with the given value and calculate the result.

Step-by-Step Calculation

Let's walk through the calculation of $f(6)$ for the function $f(x) = 7x - 4$ one more time, really emphasizing each part so it sticks. We start with our function definition: $f(x) = 7x - 4$. Our goal is to find $f(6)$.

Step 1: Substitution

The first and most critical step is to substitute the value 6 for every 'x' in the expression. So, we take $f(x) = 7x - 4$ and change it to $f(6) = 7(6) - 4$. It's super important to use parentheses around the substituted value, especially when it's being multiplied or involved in more complex operations. This helps prevent errors and clearly shows what's happening.

Step 2: Perform Multiplication

Next, we follow the order of operations (PEMDAS/BODMAS). Multiplication comes before subtraction. So, we calculate $7 \times 6$. This equals 42. Our expression now looks like $f(6) = 42 - 4$.

Step 3: Perform Subtraction

Finally, we complete the calculation by performing the subtraction. We take 42 and subtract 4 from it. $42 - 4 = 38$.

Result:

Therefore, $f(6) = 38$. The output of the function $f(x)$ when the input is 6 is 38. This whole process is fundamental to understanding how functions behave and how they can be used to model real-world scenarios. Whether you're looking at a simple linear function like this one or a much more complex polynomial or trigonometric function, the process of evaluation remains consistent. It’s all about substitution and careful calculation.

Comparing with Options

We've determined that $f(6) = 38$. Now, let's compare this result with the multiple-choice options provided:

A. 48 B. 38 C. 42 D. 46

Our calculated value, 38, perfectly matches option B. This confirms that our step-by-step evaluation was correct. It's always a good practice to double-check your work, especially when multiple-choice answers are involved, as sometimes a small calculation error can lead you to the wrong answer. Understanding why the other options are incorrect can also be beneficial. For example, 42 (option C) is the result of $7 \times 6$ before subtracting 4. Option A (48) and D (46) don't directly relate to any obvious miscalculation in this specific problem, suggesting they might be distractors or results from different incorrect operations entirely. Knowing our correct answer is 38 gives us confidence in our understanding of function evaluation.

Why is Function Evaluation Important?

Guys, understanding function evaluation isn't just about passing a math test; it's a cornerstone skill that has real-world applications and is foundational for more advanced mathematics. Think about it: functions are used everywhere! In physics, they describe motion and forces. In economics, they model supply and demand. In computer science, they're used in algorithms and data structures. When you're asked to find $f(6)$, you're practicing a skill that allows you to predict outcomes or understand behavior based on given rules. For instance, if a function describes the growth of a plant over time, evaluating it at $t=6$ (where $t$ is time) would tell you the plant's height after 6 days. If a function represents the cost of producing widgets, evaluating it with a specific number of widgets (say, 6) would tell you the total cost. This ability to plug in values and get meaningful outputs is what makes functions so powerful. It's the bridge between abstract mathematical concepts and tangible, understandable results. Master this, and you're setting yourself up for success in many different fields that rely on mathematical modeling and problem-solving. It’s the practical application of abstract thought, making math a tool rather than just a subject.

Real-World Connections

Let's connect this back to the real world. Imagine you're working at a pizza place, and the cost of making pizzas is given by a function. Let's say the function is $C(p) = 2p + 5$, where $C(p)$ is the cost in dollars and $p$ is the number of pizzas. If the manager asks, "What's the cost to make 6 pizzas?", they're essentially asking you to find $C(6)$. Using the same process we just learned: $C(6) = 2(6) + 5 = 12 + 5 = 17$. So, it costs $17 to make 6 pizzas. Or consider a scenario where you're saving money. If your savings account grows according to a function $S(m) = 10m + 50$, where $S(m)$ is your total savings after $m$ months, and you want to know how much you'll have after 6 months, you'd calculate $S(6) = 10(6) + 50 = 60 + 50 = 110$. You'd have $110 after 6 months. These are simple examples, but they illustrate how function evaluation is used to make predictions and understand financial or operational aspects in everyday life and business. It’s about translating a rule into a concrete prediction or outcome.

Conclusion

So there you have it, guys! We've successfully navigated the process of evaluating a function, specifically finding $f(6)$ for the function $f(x) = 7x - 4$. The key takeaway is that function evaluation is all about substitution and careful calculation. By replacing the variable 'x' with the given input value (in this case, 6) and then following the order of operations, we arrived at the correct answer, which is 38. We also saw how this fundamental skill connects to various real-world applications, from business costs to personal savings. Keep practicing these types of problems, as they build a strong foundation for tackling more complex mathematical concepts. Remember, math is a journey, and mastering these basics is a crucial step. You've got this!

Practice Makes Perfect

To truly solidify your understanding, the best thing you can do is practice. Try evaluating the function $f(x) = 7x - 4$ for different values of x. What is $f(0)$? What is $f(-2)$? What about $f(10)$? Each time, follow the same steps: substitute, multiply, then subtract. You can also try creating your own simple functions and evaluating them. The more you practice, the more intuitive function notation and evaluation will become. Don't be afraid to make mistakes; they are part of the learning process. The goal is to build confidence and proficiency. So, grab a piece of paper, maybe a calculator if you need it, and start plugging in those numbers! You'll be a function-evaluating pro in no time. Keep learning, keep practicing, and keep exploring the amazing world of mathematics!