Function Evaluation: Find (h+k)(2), (h-k)(3), 3h(2)+2k(3)

by Andrew McMorgan 58 views

Hey Plastik Magazine readers! Today, we're diving into the world of function evaluation with a super fun example. We're going to break down how to work with different function operations and evaluations, so you'll be a pro in no time. Get ready to sharpen your pencils and flex those math muscles, because we're about to solve a really cool problem together! If you've ever felt a little lost when it comes to functions, don't worry – we're here to make it crystal clear. Think of functions as little machines: you feed them a number, and they spit out a different number based on a set of rules. We will tackle evaluating functions, adding and subtracting them, and combining these operations. The key to mastering function evaluation lies in understanding the notation and the order of operations. Remember, math isn't just about numbers; it's about patterns and problem-solving. As we dive into this example, pay close attention to how each step builds upon the previous one. This is a common theme in mathematics, and recognizing these connections will make you a more confident and capable problem solver. This is something that you'll encounter again and again in your math journey, so it's a fantastic foundation to build. So, without further ado, let's jump right in and see how it's done!

Problem Setup: Defining Our Functions

Okay, let's get started! We're given two functions:

  • h(x) = x² + 1
  • k(x) = x - 2

These functions are the building blocks of our problem. Think of h(x) as a machine that squares whatever you put in (x) and then adds 1. Similarly, k(x) takes your input (x) and subtracts 2 from it. Before we even start calculating, it’s crucial to grasp what these functions mean. The notation might seem intimidating at first, but it's really just a shorthand way of describing a process. The most important thing to understand about functions is that they are relationships between inputs and outputs. In this case, the input is represented by the variable x, and the output is the value of the function at that x. For instance, h(x) tells us what to do with x to get a certain output, and k(x) tells us to do something different with the same input. So, when we talk about h(2), we're asking what output we get when we put 2 into the h machine. We need to understand how to manipulate functions – adding them, subtracting them, and evaluating them at specific points. This is a fundamental skill in algebra and calculus, so getting a good grasp of these concepts now will set you up for success later on. Mastering this notation and these operations is essential for anyone looking to delve deeper into mathematics and its applications. So, let's make sure we're rock solid on these basics before we move on!

Part 1: Evaluating (h + k)(2)

First up, we need to find (h + k)(2). This notation means we're adding the two functions together first, and then plugging in x = 2. Let's break it down step-by-step:

  1. Add the functions: (h + k)(x) = h(x) + k(x) = (x² + 1) + (x - 2) = x² + x - 1

    Adding functions is just like combining like terms. We take the expression for h(x) and the expression for k(x), and we add them together. Be careful to combine only the terms that have the same variable and exponent. So, we add the x² terms, the x terms, and the constant terms separately. This gives us a brand-new function, (h + k)(x), which represents the combined effect of h(x) and k(x). Notice how we've created a single function that represents the sum of the two original functions. This is a powerful technique because it allows us to treat the combined behavior of h and k as a single entity. We're essentially creating a new “machine” that performs the combined operation of the two original machines. Once we have this combined function, it's much easier to evaluate it at specific points, like x = 2 in this case. The parentheses in (h + k)(x) are important because they indicate that the addition is done before the evaluation. This is consistent with the order of operations in mathematics, which tells us to perform operations inside parentheses first. The result, x² + x - 1, is a new quadratic function. We can now treat this as a single entity and evaluate it at the desired value of x. It’s important to always simplify the expression as much as possible before substituting the value of x, because this will make the calculation easier and less prone to errors.

  2. Substitute x = 2: (h + k)(2) = (2)² + 2 - 1 = 4 + 2 - 1 = 5

    Now that we have our combined function, (h + k)(x) = x² + x - 1, we simply plug in x = 2. Remember, all we're doing here is replacing every instance of x in our expression with the number 2. This is the core idea behind evaluating a function – we're finding the output of the function for a specific input. Be careful to follow the order of operations (PEMDAS/BODMAS) when evaluating. In this case, we square 2 first (2² = 4), then we add 2, and finally we subtract 1. It's always a good idea to double-check your calculations to make sure you haven't made any arithmetic errors. A small mistake in one step can throw off the entire result. It’s also helpful to think about what this result means in the context of the original functions. We’ve found that when we combine the operations of h(x) and k(x) and then input 2, the output is 5. This gives us a concrete understanding of how the functions interact. Therefore, guys, the value of (h + k)(2) is 5. We've successfully evaluated our first expression! Let’s make sure we fully understand the steps involved: first, we added the functions h(x) and k(x) to get (h + k)(x); then, we substituted x = 2 into this new function to find the value of (h + k)(2). Each step is crucial, and understanding the logic behind each one is what makes this problem solvable. Keep these steps in mind as we tackle the next part of the problem.

Part 2: Evaluating (h - k)(3)

Next up, let's tackle (h - k)(3). Similar to before, we're subtracting the functions first and then plugging in x = 3.

  1. Subtract the functions: (h - k)(x) = h(x) - k(x) = (x² + 1) - (x - 2) = x² + 1 - x + 2 = x² - x + 3

    Subtracting functions is very similar to adding them, but there's one crucial difference: we need to be careful with the negative signs. When we subtract (x - 2), we're actually subtracting both x and -2. This means we need to distribute the negative sign to each term inside the parentheses. So, -(x - 2) becomes -x + 2. This is a common place where errors can occur, so always double-check your work when subtracting functions. Once we've distributed the negative sign, we combine like terms just as we did when adding functions. We have an x² term, an x term (-x), and constant terms (1 + 2 = 3). The result is a new function, (h - k)(x) = x² - x + 3. This function represents the difference between the outputs of h(x) and k(x) for any given x. It's important to understand that the order of subtraction matters. (h - k)(x) is not the same as (k - h)(x). If we had subtracted in the opposite order, we would have gotten a different result. This is because subtraction is not commutative, meaning that changing the order changes the answer. Now that we have the simplified expression for (h - k)(x), we can easily evaluate it at x = 3. This is the next step in solving the problem. Remember that we’re essentially finding a new function that represents the difference between the two original functions. So this expression is like a mini-program, ready to produce an output once we give it an input, which in this case is 3.

  2. Substitute x = 3: (h - k)(3) = (3)² - 3 + 3 = 9 - 3 + 3 = 9

    Now we plug in x = 3 into our simplified function, (h - k)(x) = x² - x + 3. Just like before, we replace every x with the number 3. Be sure to follow the order of operations again. First, we square 3 (3² = 9), then we subtract 3, and finally we add 3. Notice that subtracting and adding the same number cancel each other out. This simplifies our calculation and makes it easier to arrive at the correct answer. It's always a good practice to look for these kinds of simplifications, as they can save you time and reduce the chance of errors. The result, (h - k)(3) = 9, tells us that when we subtract the function k(x) from the function h(x) and then input 3, the output is 9. It's crucial to interpret these results in the context of the original problem. What does this value tell us about the relationship between the functions at this particular point? To summarize, we first found the function (h - k)(x) by subtracting k(x) from h(x), being careful to distribute the negative sign correctly. Then, we substituted x = 3 into this resulting function to find (h - k)(3). Remember how important it is to get the negative signs right – a small error there can completely change the outcome. So, guys, (h - k)(3) equals 9! We're making great progress. Only one part left!

Part 3: Evaluating 3h(2) + 2k(3)

Finally, let's calculate 3h(2) + 2k(3). This looks a little more complex, but we'll tackle it piece by piece.

  1. Evaluate h(2): h(2) = (2)² + 1 = 4 + 1 = 5

    First, we need to find the value of h(2). Remember that h(x) = x² + 1, so we simply plug in x = 2. We square 2 (2² = 4) and then add 1, giving us h(2) = 5. This is a straightforward evaluation, but it's important to get it right because it's a building block for the rest of the calculation. Always double-check your work, especially in the early stages of a problem, because errors here can propagate through the rest of the solution. By understanding how to evaluate h(2), you’re solidifying the concept of a function as a machine that takes an input (in this case, 2) and produces an output (in this case, 5). The simplicity of this calculation is deceptive – it’s a fundamental step, and understanding it perfectly sets the stage for more complex problems. Thinking of h(2) as a single value, which we've now found to be 5, is key to moving forward with the expression 3h(2) + 2k(3). We treat it like any other number, ready to be multiplied and added.

  2. Evaluate k(3): k(3) = 3 - 2 = 1

    Next, we need to find the value of k(3). We know that k(x) = x - 2, so we plug in x = 3. This gives us k(3) = 3 - 2 = 1. This is another simple evaluation, but it's crucial to get it right. Notice how we're evaluating each function independently before combining them. This is similar to the order of operations, where we perform calculations inside parentheses before doing other operations. Understanding this step reinforces the idea that functions have specific outputs for specific inputs. The function k(x), when fed the number 3, outputs 1. This is a direct application of the rule that defines k(x). Just as with h(2), we now have a single value for k(3), which we can use in the next step of the calculation. Treating k(3) as a single value, equal to 1, allows us to seamlessly incorporate it into the final expression. Keep in mind that we evaluated h(2) and k(3) separately because these are two independent parts of the expression we are evaluating. This is a common strategy in mathematics – breaking down a complex problem into smaller, more manageable parts.

  3. Substitute and calculate: 3h(2) + 2k(3) = 3(5) + 2(1) = 15 + 2 = 17

    Now that we know h(2) = 5 and k(3) = 1, we can substitute these values into our expression. We have 3h(2) + 2*k(3) = 3(5) + 2(1). We perform the multiplications first: 3 * 5 = 15 and 2 * 1 = 2. Then we add the results: 15 + 2 = 17. Remember that following the order of operations (PEMDAS/BODMAS) is crucial here. Multiplication comes before addition, so we multiply the values of the functions by their respective coefficients before adding them together. This step combines the results of our previous evaluations to give us the final answer. We've taken the individual pieces, h(2) and k(3), and combined them according to the given expression. This is a perfect example of how mathematical problems often involve multiple steps, each building upon the previous ones. Think of this as the grand finale – we’re bringing everything together to get our final answer. So, guys, 3h(2) + 2k(3) equals 17! We’ve successfully navigated a multi-part problem, demonstrating a strong understanding of function evaluation and operations. Give yourselves a pat on the back!

Final Answer

Alright, we've done it! Let's recap our answers:

  • (h + k)(2) = 5
  • (h - k)(3) = 9
  • 3h(2) + 2k(3) = 17

So there you have it, folks! We've successfully evaluated these function expressions. The key takeaways here are: break down complex problems into smaller steps, pay close attention to signs, and always follow the order of operations. You guys are now equipped to tackle similar problems with confidence. This exercise has not only helped us find the answers, but also strengthened our understanding of how functions work and how they can be manipulated. Remember, practice makes perfect! The more you work with functions, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and most importantly, keep enjoying the beauty and power of mathematics!

Remember, functions are a cornerstone of mathematics, and understanding them opens the door to more advanced topics like calculus and differential equations. Keep honing these skills, and you’ll be well on your way to mathematical mastery. Until next time, keep those brains buzzing and those pencils moving!