Finding F(m+3) For F(x) = 2/(x+5): A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun little math problem where we need to figure out what happens when we plug m+3 into the function f(x) = 2/(x+5). Don't worry, it's not as scary as it sounds! We'll break it down step-by-step so it's super easy to follow. Think of it like this: a function is like a machine, and we're just feeding it a new ingredient (m+3) instead of the usual x. Let's get started!

Understanding Function Notation

Before we jump into the calculation, let's quickly review what function notation means. The expression f(x) simply means that we have a function named "f" that takes x as an input. The function then performs some operation on x and gives us an output. In our case, the function f(x) = 2/(x+5) tells us that whatever we put in for x, we first add 5 to it, and then we divide 2 by the result. Getting comfortable with this notation is key, as it's the foundation for everything else we'll be doing. We'll be using it a lot, so if it's not quite clicking yet, don't sweat it! We'll see plenty of examples to make it crystal clear. It's like learning a new language – at first, it might seem confusing, but with a little practice, you'll be fluent in no time!

Function notation is a concise way to represent mathematical relationships and operations. It allows us to express complex ideas in a compact and understandable manner. For instance, instead of saying "take a number, add 5 to it, and then divide 2 by the result," we can simply write f(x) = 2/(x+5). This shorthand makes it much easier to work with functions and manipulate them algebraically. Moreover, function notation allows us to easily evaluate the function at different input values. For example, f(2) means we substitute x with 2 in the function's expression, giving us f(2) = 2/(2+5) = 2/7. This ability to evaluate functions at specific points is crucial in many mathematical applications, such as graphing, calculus, and optimization problems. So, mastering function notation is not just about understanding the symbols; it's about gaining a powerful tool for mathematical reasoning and problem-solving. It opens the door to a whole world of mathematical concepts and techniques. The more you practice using it, the more natural and intuitive it will become. Think of it as learning the grammar of mathematics – it's the foundation upon which you can build more complex and sophisticated ideas.

The Goal: Finding f(m+3)

Okay, so now we know what f(x) means. Our mission today is to find f(m+3). What this means is that instead of putting just x into our function, we're putting in the expression m+3. This might seem a little weird at first, but it's really just a matter of careful substitution. We're not changing the function itself; we're just giving it a different input. Imagine you have a recipe for a cake, and the recipe calls for eggs. Instead of using chicken eggs, you decide to use duck eggs. The recipe (the function) stays the same, but you're using a different ingredient (the input). That's essentially what we're doing here. We're taking the same function f(x), but instead of x, we're plugging in m+3. The goal is to simplify the expression we get after the substitution. This often involves algebraic manipulation, such as combining like terms or factoring, to arrive at the most concise and understandable form of the answer. It's like taking a complicated set of instructions and streamlining them into something much simpler and easier to follow. By finding f(m+3), we're gaining insight into how the function behaves when we give it this specific type of input. This can be useful in various applications, such as modeling real-world phenomena or solving equations. So, let's put on our detective hats and see if we can crack this case!

Remember, the key to success here is meticulous substitution. We need to make sure that we replace every instance of x in the function's expression with the expression m+3. It's like performing surgery on the function – we need to be precise and careful to avoid making any mistakes. Once we've made the substitution, we can then use our algebraic skills to simplify the result. This might involve expanding brackets, combining fractions, or any other techniques that help us to tidy up the expression. The final answer should be a simplified expression that represents the value of the function when the input is m+3. So, let's get down to business and see how it's done!

Step-by-Step Calculation

Here's where the magic happens! We'll take it slow and steady to make sure everyone's on board. Remember, our function is f(x) = 2/(x+5). To find f(m+3), we simply replace every x in the function's formula with (m+3). Think of it as a direct swap. So, wherever we see an x, we're going to put (m+3) in its place. This gives us:

f(m+3) = 2/((m+3)+5)

See? We just swapped x for (m+3). Now, the next step is to simplify the expression on the right side. We need to tidy things up a bit. Notice that we have (m+3)+5 in the denominator. We can combine the 3 and the 5 since they're just regular numbers. This is where our basic arithmetic skills come in handy. Adding these numbers together is like counting on your fingers, but instead of physical fingers, we're using our mental fingers! Once we combine those numbers, we'll have a much simpler expression in the denominator. This makes the whole fraction look a lot cleaner and easier to understand. Simplifying expressions is a fundamental skill in algebra, and it's something you'll use over and over again. It's like decluttering your room – once you get rid of all the unnecessary stuff, it's much easier to find what you need. In this case, simplifying the expression makes it easier to see the relationship between m and the function's output. So, let's roll up our sleeves and get simplifying!

This is just basic addition, guys. 3 + 5 = 8. So, our expression becomes:

f(m+3) = 2/(m+8)

And that's it! We've found f(m+3). It's the expression 2/(m+8). This means that when we plug m+3 into the function f(x), the output we get is 2 divided by m+8. Pretty neat, huh? We started with a function and a slightly strange input, and we ended up with a simple, clear expression. This is the power of algebra – it allows us to take complex problems and break them down into manageable steps. We've successfully navigated the substitution and simplification process, and we've arrived at our final answer. It's like completing a puzzle – each step fits together perfectly to reveal the final picture. In this case, the final picture is the expression 2/(m+8), which tells us exactly how the function behaves when the input is m+3. So, let's take a moment to appreciate our accomplishment and celebrate our mathematical prowess!

Final Answer

Therefore, for the function f(x) = 2/(x+5), f(m+3) = 2/(m+8). Wasn't that fun? We took a function, substituted an expression, and simplified to find our answer. This is a common type of problem in algebra, and the more you practice, the better you'll get at it. Remember the key steps: carefully substitute, simplify step-by-step, and don't be afraid to ask for help if you get stuck. We've conquered this challenge, and we're ready to tackle even more exciting math problems in the future! This journey of mathematical exploration is like climbing a mountain – each problem we solve brings us closer to the summit of understanding. And the view from the top is truly rewarding! So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. The world of mathematics is vast and full of wonders, and there's always something new to discover.

Key Takeaways

  • Substitution is key: To find f(m+3), we replaced every instance of x in f(x) with (m+3). This is a fundamental technique in algebra and calculus. It's like changing the ingredients in a recipe – you need to make sure you replace all the correct ingredients with their substitutes. In this case, we're replacing the variable x with the expression m+3. The more comfortable you become with substitution, the easier it will be to solve a wide range of mathematical problems. It's a skill that you'll use again and again throughout your mathematical journey.
  • Simplify step-by-step: We simplified the expression by combining like terms. Always take your time and work through the steps methodically to avoid errors. Think of it as building a house – you need to lay the foundation before you can put up the walls and the roof. Each step in the simplification process builds upon the previous step, so it's important to be careful and accurate. Rushing through the steps can lead to mistakes, so take a deep breath, slow down, and focus on each step individually.
  • Function notation is powerful: Understanding function notation allows us to express mathematical relationships in a concise and clear way. It's like learning a secret code – once you understand the symbols, you can unlock a whole world of mathematical ideas. Function notation is used extensively in higher-level mathematics, so mastering it now will set you up for success in the future. It's a versatile tool that can be used to represent a wide variety of functions, from simple linear functions to complex trigonometric and exponential functions.

Practice Problems

Want to test your newfound skills? Try these problems:

  1. For the function g(x) = 3x - 2, find g(a+1). Go for it, guys! This is your chance to shine and show off your amazing math skills. Remember the steps we went through together, and apply them to this new problem. It's like learning to ride a bike – the first time might be a little wobbly, but with practice, you'll be cruising along in no time. So, grab a pencil, some paper, and get ready to tackle this challenge. You've got this!
  2. If h(x) = x^2 + 4, what is h(2m)? This is another fantastic opportunity to practice your substitution and simplification skills. This problem involves a slightly different function, but the underlying principles are the same. Think carefully about how to substitute the expression 2m for x, and then simplify the result. It's like learning a new dance move – it might feel a little awkward at first, but with repetition, it will become smooth and natural. So, don't be afraid to try, make mistakes, and learn from them. That's how we grow and improve our mathematical abilities.

Keep practicing, and you'll become a function-finding pro in no time! Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, keep challenging yourself, keep exploring new concepts, and most importantly, keep having fun! The world of mathematics is full of fascinating ideas and exciting discoveries, and you're well on your way to becoming a true mathematical explorer. So, keep up the great work, and never stop learning!