Hey guys! Let's dive into a fun little math problem today. We're given a function f(x) = 5x + 40, and our mission, should we choose to accept it, is to figure out what the value of f(x) is when x equals 5. Sounds like a piece of cake, right? Well, let’s break it down and make sure we get every step crystal clear. Understanding functions is super important in math, and this is a great way to practice.

Understanding the Function

First, let's make sure we understand what a function actually is. A function is like a little machine. You feed it a number (in this case, our x value), and it spits out another number based on a specific rule. In our case, the rule is f(x) = 5x + 40. This means whatever number we put in for x, the machine will multiply it by 5 and then add 40 to the result. Simple as that! So, when we say f(5), we're asking, "What number does this machine give us when we feed it the number 5?"

Functions are the backbone of many mathematical concepts, and you'll see them everywhere from basic algebra to advanced calculus. They help us model real-world relationships. For instance, imagine you're selling lemonade. The function could represent your profit based on how many cups you sell. The x would be the number of cups, and f(x) would be your total profit after considering your costs. So, mastering functions is not just about solving equations; it’s about understanding how things relate to each other mathematically.

Another cool thing about functions is that they can be represented in many ways. We have the equation form, like f(x) = 5x + 40, but you can also represent them in a graph. If you were to graph this function, you'd get a straight line. The slope of the line would be 5, and the y-intercept would be 40. Visualizing functions can give you a better intuition for how they behave. You can also represent functions using tables, where you list different x values and their corresponding f(x) values. This can be especially helpful when you're dealing with functions that don't have a simple equation.

Substituting x = 5

Now that we've got a solid grasp of what a function is, let's get back to our problem. We need to find f(5), which means we need to substitute x with 5 in our function. So, everywhere we see an x in the equation f(x) = 5x + 40, we're going to replace it with a 5. This gives us: f(5) = 5 * 5 + 40. See how we just swapped out the x for the 5? That's the key step here.

Substitution is a fundamental technique in algebra. It allows us to evaluate expressions and solve equations by replacing variables with specific values. In many problems, you might need to substitute more complex expressions for x, but the principle remains the same. Just remember to be careful with your order of operations (PEMDAS/BODMAS) to avoid making mistakes. In our case, we need to do the multiplication before the addition, so we'll multiply 5 by 5 first, and then add 40 to the result. Getting comfortable with substitution will make solving algebraic problems much easier.

Let's consider another example to solidify the concept of substitution. Suppose we have a function g(x) = x^2 - 3x + 2, and we want to find g(2). We would substitute x with 2, so g(2) = 2^2 - 3 * 2 + 2. Then, we would evaluate the expression: g(2) = 4 - 6 + 2 = 0. This shows how substitution works even with more complex functions involving exponents and multiple terms. Practice with different functions and values to master this technique.

Performing the Calculation

Alright, now let's crunch those numbers. We've got f(5) = 5 * 5 + 40. First, we do the multiplication: 5 times 5 equals 25. So now we have f(5) = 25 + 40. Next, we add 25 and 40, which gives us 65. Therefore, f(5) = 65. Woo-hoo! We did it! This means that when we put 5 into our function machine, it spits out the number 65.

Remember that order of operations is crucial. Always do multiplication and division before addition and subtraction. If there are parentheses, solve what’s inside them first. This ensures that you get the correct answer every time. Calculators can be helpful, but it’s also good to practice doing these calculations by hand to improve your mental math skills. This can be particularly useful in situations where you don't have access to a calculator, such as during an exam. Plus, it helps you develop a better understanding of how numbers work together.

Let's try another quick calculation to reinforce these skills. Imagine we have h(x) = 10x - 15, and we want to find h(3). We substitute x with 3: h(3) = 10 * 3 - 15. First, we multiply 10 by 3, which gives us 30. Then, we subtract 15 from 30, resulting in 15. So, h(3) = 15. Keep practicing with different functions and values, and you'll become a pro at these calculations in no time.

Final Answer

So, to recap, we were given the function f(x) = 5x + 40, and we needed to find the value of f(x) when x = 5. We substituted x with 5, which gave us f(5) = 5 * 5 + 40. We then performed the calculation, which resulted in f(5) = 65. Therefore, our final answer is 65. Easy peasy, right?

Understanding how to evaluate functions is a fundamental skill in mathematics. It's used in various fields, including physics, engineering, economics, and computer science. By mastering this skill, you'll be well-equipped to tackle more complex mathematical problems. Keep practicing, and don't be afraid to ask questions if you get stuck. Math can be challenging, but it's also incredibly rewarding when you finally understand a concept.

And that’s a wrap, folks! We've successfully evaluated our function. Remember, math is all about practice, so keep at it, and you'll be a math whiz in no time! Whether you're calculating profits, modeling physical phenomena, or just doing your homework, understanding functions will be super beneficial. Keep learning, keep practicing, and most importantly, keep having fun with math!