Solving For X: When Does G(x) Equal F(2)?

Have you ever found yourself scratching your head over function problems? Don't worry, you're not alone! Let's break down a common type of problem in mathematics: finding the value of x when two functions, f(x) and g(x), are related in a specific way. In this article, we'll tackle the question: Given f(x) = 3x - 1 and g(x) = 2x - 3, for which value of x does g(x) = f(2)? We'll walk through the solution step-by-step, making sure you understand the logic behind each move. By the end, you'll be able to approach similar problems with confidence. So, let's dive in and unravel this mathematical puzzle together!

Understanding the Problem: Setting the Stage for Success

Before we jump into the calculations, let's make sure we truly grasp what the problem is asking. We are given two functions: f(x) = 3x - 1 and g(x) = 2x - 3. Think of these functions as machines: you put a number (x) in, and the machine spits out another number based on the function's rule. The question asks us to find the specific value of x that makes g(x) equal to f(2). This means we first need to figure out what f(2) is. We're not just solving a simple equation; we're comparing the outputs of two different functions. Understanding this relationship is key to finding the correct solution. The expression f(2) tells us to input the value 2 into the function f(x). Once we've calculated f(2), we'll then set g(x) equal to that value and solve for x. This two-step process—first evaluating f(2) and then solving the equation g(x) = f(2)—is the core of the problem. Visualizing the functions as machines can be helpful: we're looking for the input x for the g(x) machine that produces the same output as the f(x) machine when it receives the input 2. This conceptual understanding makes the algebraic steps much more meaningful and less like just memorizing a procedure. So, let's get ready to put on our mathematical thinking caps and solve this puzzle!

Step-by-Step Solution: Unraveling the Mathematical Puzzle

Now, let's break down the solution into clear, manageable steps. This step-by-step approach will not only help us find the answer but also solidify our understanding of the process. Remember, the goal is not just to get the right answer but to learn how to get the right answer. So, let's take our time and focus on each step. First, we need to calculate f(2). This is the starting point of our journey. We'll substitute x with 2 in the function f(x) = 3x - 1. This gives us f(2) = 3(2) - 1. Now, we simply perform the arithmetic: 3 multiplied by 2 is 6, and 6 minus 1 is 5. Therefore, f(2) = 5. This means that when we input 2 into the function f(x), the output is 5. Next, we'll use this value to solve for x in the equation g(x) = f(2). Since we know f(2) = 5, we can rewrite the equation as g(x) = 5. Now, we substitute the expression for g(x), which is 2x - 3, into the equation. This gives us 2x - 3 = 5. We now have a simple linear equation to solve for x. To isolate x, we first add 3 to both sides of the equation: 2x - 3 + 3 = 5 + 3, which simplifies to 2x = 8. Finally, we divide both sides of the equation by 2: 2x / 2 = 8 / 2, which gives us x = 4. Therefore, the value of x for which g(x) = f(2) is 4. We have successfully navigated through the problem, step by step, and arrived at our solution!

Detailed Solution: Showcasing the Mathematical Steps

Let's formalize the solution by presenting it in a clear, mathematical format. This will reinforce our understanding and provide a template for solving similar problems in the future. We'll start by restating the given functions: f(x) = 3x - 1 and g(x) = 2x - 3. Our goal is to find the value of x such that g(x) = f(2). First, we evaluate f(2) by substituting x = 2 into the function f(x):

f(2) = 3(2) - 1

Performing the multiplication, we get:

f(2) = 6 - 1

Subtracting 1 from 6, we find:

f(2) = 5

Now that we know f(2) = 5, we can set g(x) equal to 5 and solve for x:

g(x) = f(2)

g(x) = 5

We substitute the expression for g(x), which is 2x - 3:

2x - 3 = 5

To isolate x, we add 3 to both sides of the equation:

2x - 3 + 3 = 5 + 3

2x = 8

Finally, we divide both sides of the equation by 2:

2x / 2 = 8 / 2

x = 4

Thus, the value of x for which g(x) = f(2) is 4. This detailed solution showcases each step in the process, making it easy to follow and understand. By presenting the solution in this format, we can clearly see the mathematical logic and reasoning behind each step.

Why This Matters: Real-World Applications and Beyond

You might be wondering, "Why is this type of problem important?" While it might seem abstract, understanding functions and how they relate to each other is crucial in many areas of mathematics and real-world applications. Functions are the building blocks of more advanced mathematical concepts like calculus, which is used extensively in physics, engineering, economics, and computer science. The ability to manipulate functions, solve equations involving functions, and understand their behavior is a fundamental skill. In fields like engineering, functions are used to model systems and predict their behavior. For example, engineers might use functions to model the trajectory of a rocket, the flow of electricity in a circuit, or the stresses in a bridge. By understanding how functions work, they can design better systems and solve complex problems. In economics, functions are used to model supply and demand, cost and revenue, and other economic relationships. Economists can use these models to make predictions about the economy and to design policies that promote economic growth. In computer science, functions are the cornerstone of programming. Programs are built from functions that perform specific tasks, and understanding how functions work is essential for writing effective code. Even in everyday life, we encounter functions all the time. Think about the relationship between the number of hours you work and the amount of money you earn, or the relationship between the amount of fuel you put in your car and the distance you can travel. These are all examples of functions in action. So, by mastering the basics of functions, you're not just learning math; you're developing skills that will be valuable in a wide range of fields and in your everyday life.

Conclusion: Mastering Functions for Mathematical Success

We've successfully navigated through the problem of finding the value of x for which g(x) = f(2), given the functions f(x) = 3x - 1 and g(x) = 2x - 3. We broke down the problem into manageable steps, evaluated f(2), set up the equation g(x) = f(2), and solved for x. Along the way, we emphasized the importance of understanding the problem conceptually and presented a detailed solution to solidify our understanding. Remember, practice makes perfect! The more you work with functions and solve problems like this, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're a natural part of the learning process. Each time you encounter a new problem, take a deep breath, break it down into smaller steps, and apply the concepts you've learned. And remember, understanding functions is not just about getting the right answer; it's about developing a deeper understanding of mathematics and its applications in the world around us. So, keep practicing, keep exploring, and keep learning! For further exploration and practice with functions, consider visiting resources like Khan Academy's Functions and Equations section. They offer a wealth of tutorials, exercises, and videos to help you master this essential mathematical concept.