Evaluating (m/n)(x) For X = -3 A Mathematical Exploration

Hey everyone! Today, we're diving into a fun little math problem where we need to evaluate a function, specifically extit{(m/n)(x)}, at a particular point, x = -3. We're given two functions, m(x) = x² + 4x and n(x) = x. Sounds interesting? Let's break it down together, step by step, so it's super clear. Understanding these kinds of problems is crucial for anyone delving into algebra and calculus, as they form the building blocks for more complex concepts. So, grab your calculators (or just your brainpower!) and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we fully grasp what the problem is asking. We have two functions: extbf{m(x)}, which is a quadratic function (x² + 4x), and extbf{n(x)}, which is a simple linear function (x). The notation extit{(m/n)(x)} represents a new function formed by dividing m(x) by n(x). Essentially, we're creating a rational function. Our ultimate goal is to find the value of this combined function when x is -3. This means we'll substitute -3 for every 'x' in our expression and simplify. It's like following a recipe – we have the ingredients (functions) and the instructions (evaluate at x = -3), and we just need to put them together correctly. This type of problem is common in algebra, and it tests our understanding of function notation, substitution, and simplification. The key here is to be methodical and pay attention to detail, especially when dealing with negative numbers. We don't want any pesky sign errors throwing us off! Think of it like this: if m(x) represents the total cost of something depending on the quantity x, and n(x) represents the number of items, then (m/n)(x) could represent the cost per item. Evaluating it at x = -3, while not practically meaningful in this context (you can't have a negative number of items!), helps us understand the mathematical behavior of the function. So, let's move on to the first step: forming the (m/n)(x) function.

Step 1: Forming the (m/n)(x) Function

Okay, so the first thing we need to do is actually create the extit{(m/n)(x)} function. Remember, this just means dividing m(x) by n(x). We know that m(x) is x² + 4x and n(x) is x. So, (m/n)(x) looks like this: (x² + 4x) / x. Now, before we go plugging in x = -3, let's see if we can simplify this expression a bit. Simplification often makes the evaluation step easier and less prone to errors. Looking at the numerator (x² + 4x), we can see that both terms have a common factor of 'x'. We can factor out an 'x' from the numerator: x(x + 4). So now our (m/n)(x) function looks like this: x(x + 4) / x. Ah, a beautiful simplification is about to happen! We have an 'x' in both the numerator and the denominator. As long as x isn't zero (we'll talk about why that's important in a bit), we can cancel these out. This leaves us with a much simpler expression: (m/n)(x) = x + 4. This is a crucial step! We've transformed a potentially messy rational function into a simple linear function. This makes our evaluation in the next step a piece of cake. But, before we move on, let's quickly address why we mentioned x not being zero. Remember, you can't divide by zero in mathematics. It's a big no-no! So, while we simplified our function, we need to keep in mind that our original function, (x² + 4x) / x, is undefined when x = 0. This is called a restriction on the domain of the function. Even though our simplified function, x + 4, is perfectly happy at x = 0, the original function isn't. This is a subtle but important point in understanding rational functions. Now that we have our simplified function, (m/n)(x) = x + 4, we're ready to substitute x = -3.

Step 2: Substituting x = -3

Alright, we've got our simplified function, extit{(m/n)(x) = x + 4}, and we're ready to plug in x = -3. This step is pretty straightforward. We just replace every 'x' in our function with '-3'. So, (m/n)(-3) becomes -3 + 4. This is a simple addition problem. -3 + 4 equals 1. So, (m/n)(-3) = 1. And that's it! We've evaluated our function at x = -3. This step highlights the power of simplification. If we had tried to plug -3 into the original function, (x² + 4x) / x, we would have gotten (-3² + 4(-3)) / -3, which simplifies to (9 - 12) / -3, which is -3 / -3, which also equals 1. But, plugging into the simplified function was much easier and less prone to error. This also demonstrates why understanding the order of operations is crucial. We perform the operations inside the parentheses first (if any), then exponents, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). In our case, we simply had an addition operation to perform. The result, (m/n)(-3) = 1, is a single numerical value. It represents the output of the function (m/n)(x) when the input is -3. We can think of this as a point on the graph of the function (m/n)(x). The point would be (-3, 1). So, we've not only found the value but also gained some insight into the function's behavior. Now, let's take a moment to recap our steps and discuss the significance of our answer.

Step 3: The Final Answer and its Significance

Okay, let's recap what we've done. We started with two functions, m(x) = x² + 4x and n(x) = x, and we needed to evaluate (m/n)(x) at x = -3. First, we formed the (m/n)(x) function by dividing m(x) by n(x), resulting in (x² + 4x) / x. Then, we simplified this expression by factoring out an 'x' from the numerator and canceling it with the 'x' in the denominator, giving us (m/n)(x) = x + 4. Finally, we substituted x = -3 into our simplified function, resulting in (m/n)(-3) = -3 + 4 = 1. So, our final answer is 1. But what does this answer really mean? Well, it tells us the value of the combined function (m/n)(x) at the specific point x = -3. In graphical terms, it means that the point (-3, 1) lies on the graph of the function (m/n)(x). More broadly, this exercise demonstrates several important concepts in algebra: function notation, forming new functions by combining existing ones, simplifying expressions, substitution, and evaluating functions at specific points. These are fundamental skills that you'll use again and again in more advanced math courses. It also highlights the importance of paying attention to detail and being methodical in your approach. A single sign error or missed step can lead to a wrong answer. Furthermore, we touched upon the concept of domain restrictions. While our simplified function, x + 4, is defined for all real numbers, the original function, (x² + 4x) / x, is undefined at x = 0. This is a crucial point to remember when working with rational functions. Understanding these nuances will help you avoid common pitfalls and develop a deeper understanding of mathematical concepts. So, we've not just solved a problem; we've reinforced some essential mathematical skills and concepts. Keep practicing, and you'll become a math whiz in no time!

So, there you have it, guys! We've successfully evaluated extit{(m/n)(x)} for x = -3 and found the answer to be 1. We walked through each step carefully, from understanding the problem to forming the function, simplifying it, substituting the value, and finally, interpreting the result. Remember, the key to mastering these types of problems is practice and a solid understanding of the underlying concepts. Don't be afraid to break down complex problems into smaller, more manageable steps. And always double-check your work to avoid those sneaky little errors! Keep exploring, keep learning, and keep having fun with math! You've got this!