Hey guys! Let's dive into a super straightforward math problem today. We're given a function, f(x) = 5x + 40, and our mission is to find out what f(x) equals when x is 5. Sounds like a piece of cake, right? Well, it is! This type of problem is a staple in algebra, and mastering it will definitely boost your confidence. So, grab your imaginary pencils, and let's get started!

Understanding the Function

Before we jump into plugging in numbers, let's break down what the function f(x) = 5x + 40 actually means. In simple terms, a function is like a machine: you feed it a number (in this case, 'x'), and it spits out another number based on a specific rule. This rule is the equation itself. In our case, the rule says: "Take the input 'x', multiply it by 5, and then add 40." Functions are the building blocks of more advanced math, so getting comfortable with them now is super important. Recognizing the structure of a function helps you understand how different inputs affect the output. This understanding is crucial not just for solving problems but also for grasping the underlying mathematical concepts. Think of functions as a way to transform numbers – they're like little number-manipulating wizards! When you see 'f(x)', just remember it's a way of saying "the value of the function f at the point x". It’s a notation that tells us we’re dealing with a relationship between an input (x) and an output (f(x)). And remember, functions aren't limited to simple algebraic expressions like the one we have here; they can get much more complex, involving trigonometric, logarithmic, or even piecewise definitions. But the fundamental idea remains the same: input, process, output.

Plugging in the Value

Alright, now for the fun part: plugging in the value! We're asked to find f(5), which means we need to replace every 'x' in our function with the number 5. So, our equation becomes f(5) = 5 * 5 + 40. See? We've simply swapped 'x' with '5'. This is the core of evaluating functions at specific points. By substituting the variable with a numerical value, we can determine the function's output at that particular input. It's like giving the function a specific instruction: "Hey, calculate your value when x is 5!" This process is fundamental in many areas of mathematics and its applications, from graphing functions to solving equations. The act of substitution allows us to move from a general expression, valid for any value of x, to a specific numerical result. It bridges the gap between abstract algebraic representation and concrete numerical calculation. And as you progress in your mathematical journey, you'll find that the principle of substitution is applied in countless contexts, making it an indispensable skill. So, remember: whenever you're asked to evaluate a function at a certain point, just replace the variable with the given value and simplify!

Solving the Equation

Now that we have f(5) = 5 * 5 + 40, it's just a matter of following the order of operations (PEMDAS/BODMAS). First, we multiply: 5 * 5 = 25. Then, we add: 25 + 40 = 65. Boom! We've got our answer. So, f(5) = 65. This step highlights the importance of following the correct order of operations. Multiplication comes before addition, and adhering to this rule ensures that we arrive at the correct answer. It's a simple yet crucial aspect of mathematical calculations. The order of operations isn't just an arbitrary convention; it's a fundamental principle that ensures consistency and avoids ambiguity in mathematical expressions. Without it, different people could interpret the same expression in different ways, leading to chaos. So, always remember to prioritize operations according to the PEMDAS/BODMAS rule: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Mastering this order is essential for accurate and reliable mathematical problem-solving. Think of it as the grammar of mathematics – it ensures that our expressions are clear, unambiguous, and universally understood.

The Answer

Therefore, if f(x) = 5x + 40, then f(5) = 65. Wasn't that easy? You see how straightforward it can be once you understand the basic concept. Always remember to break down the problem into smaller, manageable steps. This approach makes even complex problems seem less daunting. By systematically addressing each step, you increase your chances of arriving at the correct solution. And don't be afraid to practice! The more you work through different types of problems, the more comfortable you'll become with the underlying principles. Math is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing, keep exploring, and keep challenging yourself. With consistent effort, you'll be amazed at how much progress you can make. And remember, every problem you solve is a step forward on your mathematical journey!

Real-World Applications

You might be wondering, "Okay, that's cool, but where would I ever use this in real life?" Well, functions are everywhere! They're used to model all sorts of relationships. For example, imagine you're calculating the cost of a taxi ride. The total cost might be a function of the distance traveled, with a base fare plus a per-mile charge. Or, think about calculating your earnings based on an hourly wage plus tips. The total earnings would be a function of the hours worked and the amount of tips received. These are just a couple of examples, but the possibilities are endless. Functions are also essential in computer programming, where they're used to create reusable blocks of code that perform specific tasks. In science, functions are used to model physical phenomena, such as the trajectory of a projectile or the growth of a population. Even in economics, functions are used to analyze supply and demand, predict market trends, and optimize resource allocation. The ability to understand and work with functions is a valuable skill that can be applied in a wide range of fields. So, the next time you're faced with a real-world problem, think about whether a function could be used to model the relationship between different variables. You might be surprised at how often they come in handy!

Practice Problems

Want to test your understanding? Try these practice problems:

1. If g(x) = 2x - 7, find g(3).
2. If h(x) = x^2 + 1, find h(4).
3. If k(x) = (x + 5) / 2, find k(1).

Solving these problems will solidify your understanding of how to evaluate functions. Remember to follow the same steps we used in the example problem: identify the function, substitute the given value for x, and simplify the expression. And don't be afraid to make mistakes! Mistakes are a natural part of the learning process. The key is to learn from your mistakes and keep practicing. If you get stuck, review the steps we discussed earlier or seek help from a teacher, tutor, or online resource. There are plenty of resources available to help you succeed in mathematics. And with a little bit of effort, you'll be able to master the art of evaluating functions and apply this skill to solve a wide range of problems. So, grab a pencil and paper, and get ready to put your newfound knowledge to the test! The more you practice, the more confident you'll become in your mathematical abilities.

Conclusion

And there you have it! Evaluating functions is a fundamental skill in algebra with tons of applications. Keep practicing, and you'll be a pro in no time! Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts and applying them to solve problems. By focusing on understanding, you'll be able to tackle even the most challenging problems with confidence. And don't be afraid to ask questions! Asking questions is a sign of strength, not weakness. It shows that you're engaged in the learning process and that you're willing to seek help when you need it. So, keep asking questions, keep exploring, and keep learning. The world of mathematics is vast and fascinating, and there's always something new to discover. So, embrace the challenge, and enjoy the journey!

Keep up the great work, guys! You've got this! Always remember to practice and understand the core concepts, and you'll be able to solve a lot more complex problems with ease. Good luck!