Finding (f+g)(2): A Step-by-Step Math Guide

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Hey guys! Let's dive into a cool math problem. We're gonna find the value of (f+g)(2). This kind of problem often pops up in algebra, and it's super important to understand the basics. Don't worry, it's not as scary as it looks. We'll break it down into easy-to-follow steps. If you're struggling with similar problems, you're in the right place! We'll cover everything you need to know. Remember, the key is to stay patient and practice. Ready? Let's go!

Understanding the Problem: What Does (f+g)(2) Mean?

Okay, before we start crunching numbers, let's get clear on what (f+g)(2) actually means. In math, when we see something like this, it tells us we need to do a couple of things. First, we need to add the functions f(x) and g(x) together. Think of it like combining two recipes to make a new dish. Then, once we have that combined function, we'll substitute the value '2' wherever we see 'x'. This is like saying, “Hey, in this new recipe, let’s use 2 as the ingredient.” Simple, right? Essentially, (f+g)(2) is asking us: “What's the output when we plug in 2 into the combined function of f(x) and g(x)?” Understanding this is crucial before jumping into the calculations. If you're a beginner, just remember this: combine the functions first, then plug in the number.

So, in our specific problem, we're given:

  • f(x) = 2 + x²
  • g(x) = 5x + 5

Our task is to find (f+g)(2). This will become easy peasy once we start the calculations. Let's get to it!

Step-by-step Explanation

Now, let's crack on with the calculation. It's like a fun puzzle that we need to solve step by step. Just follow along, and you'll be acing these problems in no time. Take it easy, there's no rush. Let's solve it step by step:

  1. Combine the functions f(x) and g(x): This means we add the two functions together. We have f(x) = 2 + x² and g(x) = 5x + 5. So, (f + g)(x) = (2 + x²) + (5x + 5). Now, let's simplify this. We combine like terms (the numbers and the terms with 'x').

    • (f + g)(x) = x² + 5x + 7

  2. Substitute x = 2 into the combined function: Now that we have our combined function, we're ready to plug in '2' wherever we see 'x'. So, we replace every 'x' in the expression x² + 5x + 7 with '2'.

    • (f + g)(2) = (2)² + 5(2) + 7

  3. Calculate the result: Let's finish this off! Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). First, calculate the exponent (2² = 4), then do the multiplication (5 * 2 = 10). Finally, add everything up.

    • (f + g)(2) = 4 + 10 + 7
    • (f + g)(2) = 21

So, there you have it! The value of (f+g)(2) is 21. See? It wasn't that hard after all! Just a matter of taking things slowly and carefully.

Tips and Tricks for Similar Problems

Alright, now that we've nailed this one, let's get you ready to crush any similar problems that come your way. Here are some pro tips and tricks that will help you excel:

  • Practice, practice, practice: The more problems you solve, the better you'll become. Start with easier problems and gradually move to more complex ones.
  • Understand the basics: Make sure you're comfortable with the fundamental concepts of functions, substitution, and combining like terms. If you struggle with the basic principles, you can look for more explanations and examples.
  • Use the order of operations: Always follow PEMDAS/BODMAS to avoid calculation errors. This helps to ensure you don't make any silly mistakes.
  • Write everything down: Don't try to do too much in your head, especially when you're just starting out. Writing down each step helps prevent errors and allows you to track your process.
  • Check your work: After you've found your answer, go back and double-check your calculations. This can help you catch any mistakes you might have made.
  • Break it down: When you encounter a complicated problem, break it down into smaller, more manageable steps. This makes the problem less daunting and easier to solve.
  • Be patient: Don't get discouraged if you don't understand something right away. Math takes time and practice. Keep at it, and you'll get there!

Common Mistakes to Avoid

Okay, guys, let's talk about some common pitfalls to avoid. Knowing these will save you a lot of headache in the long run.

  • Mixing up the order of operations: Always, always follow PEMDAS/BODMAS. It's easy to make mistakes if you don't do this.
  • Incorrectly substituting values: Be careful when substituting numbers into the functions. Make sure you replace every 'x' with the correct value and keep track of parentheses.
  • Forgetting to combine like terms: This is a very common mistake. Make sure you combine all the similar terms (constants and terms with the same variable and exponent) correctly.
  • Not simplifying properly: Always simplify your final answer as much as possible. This helps you get the most accurate result.
  • Misunderstanding the question: Ensure you fully understand what the question is asking before you start solving. Read the problem carefully and make sure you know what you're looking for.

By keeping these tips in mind, you'll be well-prepared to tackle any similar problem that comes your way. Remember, it's all about practice and understanding the basics!

Further Practice Problems

Ready to get some more practice? Awesome! Here are a few more problems to test your skills:

  1. Given f(x) = 3x - 1 and g(x) = x² + 4, find (f + g)(1).
  2. If f(x) = x² - 2x and g(x) = 2x + 1, what is (f - g)(3)?
  3. If f(x) = 4x + 2 and g(x) = x - 3, find (f * g)(0).

Try these problems on your own, and then check your answers. This will give you more confidence when dealing with these kinds of questions. Don't worry if you get stuck; just go back to the steps we used earlier and review any parts that you didn't quite understand. Remember that every problem you solve makes you better and more prepared!

Conclusion: You've Got This!

Alright, we've reached the end, guys! You now have a solid understanding of how to find the value of (f+g)(2). You've seen the step-by-step process, learned some helpful tips, and discovered some common mistakes to avoid. Remember, learning math is a journey, not a race. So, don't get discouraged if you don't get it right away. Keep practicing, stay curious, and keep asking questions. You've got this! Math can be fun if you break it down into manageable steps. Now go out there and keep those math muscles flexed!

Keep up the great work, and good luck with your future math adventures!