Hey guys! Let's dive into a classic math problem. We're given a function, f(x) = 5x + 40, and our mission, should we choose to accept it, is to figure out what f(5) is. Sounds easy, right? It totally is! This is a great example of how functions work and how to substitute values into them. It’s like a little puzzle, and we have all the pieces.

Understanding the Function: The Basics

First off, let's break down what a function actually is. Think of a function like a machine. You put something in (in this case, an x value), the machine does something to it (multiplies it by 5 and adds 40), and out pops a result (the f(x) value). The f(x) = 5x + 40 equation is the rule, the instructions, or the program for this machine. It tells us exactly what to do with any number we put in for x. The beauty of functions is that they give us a way to describe relationships between numbers. Functions are everywhere in mathematics and in computer programming, making them a fundamental concept. They let us model and understand how different things relate to each other. Understanding functions is a stepping stone to more complex mathematical concepts.

In our particular function, f(x) = 5x + 40, the 'x' is the input. It's a placeholder. We can put any number we want in place of 'x'. Then, the function tells us to multiply that number by 5 and then add 40. The result of that calculation is the output, or f(x). For example, if we put in x = 1, the function tells us to calculate (5 * 1) + 40, which equals 45. So, f(1) = 45. Pretty straightforward, right? This concept of input, process, and output is consistent across all functions. You just need to know the rule (the equation) to get the answer. This is fundamental in subjects like calculus or algebra, especially when graphing functions or analyzing them. So understanding this basic step is essential.

The Substitution Step: Plugging in the Value

Now, here comes the fun part: finding f(5). This means we are going to substitute every instance of 'x' in the function with the number 5. So, instead of f(x) = 5x + 40, we’ll have f(5) = 5(5) + 40. See how we just replaced the 'x' with '5'? Easy peasy!

This substitution is a core skill in algebra and is used extensively in all the sciences. You are literally just replacing one symbol with another. In more complex problems, you might substitute entire expressions, not just single numbers, but the principle is the same. It's all about following the rule. In this case, our rule is the original equation, and the instructions are very clear: replace x with 5 and solve. This is a very common approach in mathematical problem-solving, not just in algebra but in almost all fields of mathematics. Being able to substitute values is a fundamental skill that builds your capacity to solve more complex equations. By making this simple substitution, we're setting up the problem to be solved very easily.

Performing the Calculation: The Final Answer

Now that we've substituted 5 for x, we have a simple arithmetic problem to solve: f(5) = 5(5) + 40. Let's break it down step-by-step.

First, multiply 5 by 5: 5 * 5 = 25. Next, add 40 to the result: 25 + 40 = 65. Therefore, f(5) = 65! We have found the output of the function when the input is 5. Congratulations, you’ve solved it!

The calculation phase is where you use your basic arithmetic skills to reach a final answer. In most problems, the substitution step is usually the more difficult part. Once you've completed the substitution, then it's just basic arithmetic. In our case, the calculation involves simple multiplication and addition, but other functions can involve more complex arithmetic operations. Practicing these kinds of calculations strengthens your math muscle. You should always double-check your work, particularly when it comes to the arithmetic stage, to make sure you didn’t make a careless mistake. This step reinforces the concepts of order of operations, which is crucial for consistency in your problem-solving. This step ensures that the process is applied systematically and accurately. The ability to calculate accurately and quickly is an important skill when dealing with functions.

Key Takeaways and why it matters

So, what have we learned? We’ve learned how to evaluate a function. We've replaced the variable x with a given number (5 in this case) and simplified the equation to find the value of the function at that point. We understand that functions take an input, apply a rule, and produce an output. This is a crucial skill in algebra and will be used continuously as you progress through more advanced mathematics.

This simple example illustrates the fundamentals of functions. The ability to evaluate functions is an essential skill in mathematics and in programming. This kind of problem is the foundation of more complex problems that you'll encounter in calculus, statistics, and other branches of mathematics. The ability to substitute values into equations and solve for an answer is a skill that will serve you throughout your mathematical career. Functions are used everywhere, from calculating the trajectory of a rocket to predicting stock market trends.

Further Exploration: More Examples

Let’s try a few more examples just to solidify our understanding. What if the function was f(x) = 2x - 10? What would f(3) be? Well, let’s substitute x with 3: f(3) = 2(3) - 10. Then, we multiply 2 by 3 to get 6. Then we subtract 10, which gives us f(3) = -4.

What about f(0)? Using the original function f(x) = 5x + 40, we would substitute x with 0: f(0) = 5(0) + 40. So, f(0) = 0 + 40 = 40. This illustrates that functions can output any number depending on the input. You can keep playing with different values of x to see how the output changes. Changing the value of x changes the outcome every single time. And that is what functions are all about.

Function notation

It’s also important to understand function notation in general. Function notation, which often uses f(x), g(x), or h(x), is a way to represent a relationship between an input and an output. It’s a very formal and specific way to express this relationship. The letter inside the parentheses indicates the input variable, and the entire expression represents the output based on a predefined rule. It's a universal language in mathematics.

For example, f(x) tells us that we have a function named ‘f’ that takes the input ‘x’. If we had g(x) = x^2, it tells us that our function ‘g’ takes the input ‘x’ and squares it. Similarly, if we see h(x) = sin(x), it means the function ‘h’ takes the input ‘x’ and calculates its sine. Function notation makes it simpler to write mathematical rules and define relationships between different variables. It clearly indicates that the output is related to the input by a specific rule. This concise notation is key to understanding and solving complex math problems. It also makes it easy to compare and contrast multiple functions. This notation helps to define and interpret mathematical expressions systematically.

Conclusion: You've Got This!

So, there you have it, guys! We have successfully evaluated the function f(x) = 5x + 40 when x = 5. You now have a solid understanding of how to work with functions and substitute values. Remember the steps: understand the function, substitute the value, and perform the calculation. You're now well-equipped to tackle similar problems. Keep practicing and exploring, and you'll become a function-solving pro in no time! You've got the skills to handle these types of questions. Keep up the amazing work, and keep exploring the amazing world of mathematics! You're off to a great start!