Decoding (p+q)(2) A Step-by-Step Guide

Hey guys! Today, we are diving into a fun little math problem that involves evaluating expressions using tables. We've got two functions here, p(x) and q(x), neatly laid out in their respective tables. Our mission, should we choose to accept it, is to find the value of (p+q)(2). Don't worry, it's not as daunting as it sounds! We're going to break it down piece by piece, making sure everyone's on board.

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the problem is asking. The expression (p+q)(2) might look a bit cryptic at first, but it's actually quite straightforward. It simply means we need to add the values of the functions p(x) and q(x) when x is equal to 2. In other words, we need to find p(2), find q(2), and then add those two values together. This concept is fundamental in understanding how functions work, and it's a cornerstone of many mathematical concepts you'll encounter later on. So, grasping this now will set you up for success down the road. Think of functions as little machines: you feed them a number (x), and they spit out another number (p(x) or q(x)). The tables we have are simply showing us what these machines do for specific input values. We are essentially using the tables as a lookup to find the corresponding output values for a given input.

Breaking Down the Notation

Let's clarify the notation further. The expression p(x) represents the value of the function p at the point x. Similarly, q(x) represents the value of the function q at the point x. When we see (p+q)(x), it means we are adding the values of the two functions at the same x value. So, (p+q)(x) = p(x) + q(x). In our specific case, we want to find (p+q)(2), which means we need to find p(2) and q(2) and then add them. It's like saying, “What does the p machine do when we put in 2? What does the q machine do when we put in 2? Now, let's add those results together!” This way of thinking about functions can help you visualize what's happening and make the process less abstract. Remember, math isn't about memorizing formulas; it's about understanding the underlying concepts. Once you understand the concept, the formulas become tools to help you solve problems, rather than hurdles to overcome.

Why Tables are Useful

Tables are a fantastic way to represent functions, especially when we don't have a specific algebraic formula for the function. They give us a clear and concise way to see the relationship between input (x) and output (p(x) or q(x)). In real-world applications, you might encounter data presented in tables like this all the time – think of things like weather data, stock prices, or experimental results. Learning how to read and interpret tables is a crucial skill, not just in math class, but in many different areas of life. The tables provided in our problem are particularly helpful because they directly give us the values we need to solve the problem. We don't have to do any complicated calculations or plug numbers into equations. We simply need to look up the values in the table and then perform a simple addition. This highlights the power of organization and representation in problem-solving. Sometimes, the best way to tackle a problem is to present the information in a clear and organized manner, making it easier to see the solution. So, let's get to the solution by navigating through the tables.

Finding p(2) and q(2)

The first step in solving this problem is to locate the values of p(2) and q(2) from the given tables. Let's start with p(2). Looking at the first table, we need to find the row where x is equal to 2. Once we find that row, we can read off the corresponding value of p(x). In this case, when x is 2, p(x) is 3. So, we've found that p(2) = 3. See how easy that was? The table directly gave us the answer. Now, let's do the same for q(2). We'll look at the second table and find the row where x is 2. Then, we'll read off the corresponding value of q(x). In this case, when x is 2, q(x) is -2. So, q(2) = -2. We've now successfully found both p(2) and q(2) using the tables. This step is crucial because it provides the building blocks for the final calculation. Without these values, we can't determine the value of (p+q)(2). The process of reading values from a table is a fundamental skill in data analysis and interpretation. Whether you're working with spreadsheets, scientific data, or financial reports, the ability to quickly and accurately extract information from tables is invaluable. So, practice this skill whenever you get the chance, and you'll become a master of data interpretation in no time!

Decoding the Tables

Let's delve deeper into how we extract this information from the tables. Think of each table as a map. The x values are like the addresses, and the p(x) or q(x) values are the corresponding information at those addresses. When we're asked to find p(2), it's like we're given the address