Solving For X When F(x) = G(x) Given F(x) = -3x + 4 And G(x) = 2
Introduction
In mathematics, a fundamental task involves finding the values of variables that satisfy given equations. This article delves into the process of solving for the value of x when two functions, f(x) and g(x), are equal. Specifically, we are given the functions f(x) = -3x + 4 and g(x) = 2, and our goal is to determine the x value for which f(x) = g(x). This exploration will involve algebraic manipulation and a clear understanding of function evaluation. Understanding how to solve equations formed by function equality is crucial in various fields, including physics, engineering, and economics, where mathematical models are used to describe real-world phenomena. By mastering these techniques, students and professionals alike can enhance their problem-solving skills and apply mathematical concepts to practical situations. The following sections will guide you through a step-by-step solution, ensuring clarity and comprehension at each stage. We will begin by setting up the equation, followed by isolating the variable x, and finally, verifying our solution. This methodical approach will not only provide the answer but also reinforce the underlying mathematical principles. So, let's embark on this mathematical journey and unlock the value of x that makes the equation f(x) = g(x) true.
Setting up the Equation
The initial step in solving for x is to set the two functions equal to each other. Given that f(x) = -3x + 4 and g(x) = 2, we can write the equation as follows:
-3x + 4 = 2
This equation represents the condition where the outputs of the two functions are the same. By setting up the equation in this manner, we establish a clear pathway to isolate the variable x and find its value. The significance of this step cannot be overstated, as it forms the foundation for the entire solution process. A clear and accurate setup ensures that subsequent algebraic manipulations are performed correctly, leading to the correct answer. It's essential to recognize that we are essentially finding the x-coordinate(s) where the graphs of the two functions intersect. In this specific case, f(x) is a linear function with a slope of -3 and a y-intercept of 4, while g(x) is a horizontal line at y = 2. The point of intersection, if it exists, will provide the solution to our equation. As we proceed, we will employ algebraic techniques to isolate x and determine its precise value. This methodical approach ensures that we not only arrive at the correct solution but also understand the underlying mathematical principles. The next section will detail the steps involved in isolating x, further elucidating the solution process.
Isolating the Variable x
Now that we have the equation -3x + 4 = 2, our next objective is to isolate the variable x. This involves performing algebraic operations on both sides of the equation to gradually separate x from the constants. The first step is to subtract 4 from both sides of the equation. This maintains the balance of the equation while moving the constant term to the right side:
-3x + 4 - 4 = 2 - 4
Simplifying this gives us:
-3x = -2
Next, to completely isolate x, we need to divide both sides of the equation by -3. This will remove the coefficient multiplying x and leave us with x alone on the left side:
-3x / -3 = -2 / -3
This simplifies to:
x = 2/3
Thus, we have found that the value of x that satisfies the equation f(x) = g(x) is 2/3. The process of isolating x is a fundamental skill in algebra, requiring a careful application of inverse operations to maintain the equation's balance. By performing these steps methodically, we can confidently arrive at the correct solution. It's important to note that each operation must be applied to both sides of the equation to ensure that the equality remains valid. In this case, subtracting 4 and then dividing by -3 were the necessary steps to isolate x. With the value of x determined, it's a good practice to verify the solution to ensure accuracy. The subsequent section will cover the verification process, reinforcing our understanding and confirming the correctness of our answer.
Verifying the Solution
After finding a solution, it is crucial to verify its correctness. This step involves substituting the obtained value of x back into the original functions f(x) and g(x) and confirming that f(x) = g(x). In our case, we found that x = 2/3. Let's substitute this value into f(x) = -3x + 4:
f(2/3) = -3(2/3) + 4
Simplifying this expression:
f(2/3) = -2 + 4
f(2/3) = 2
Now, let's evaluate g(x) at x = 2/3. Since g(x) = 2, the value of g(x) is constant and does not depend on x:
g(2/3) = 2
Comparing the results, we see that:
f(2/3) = 2
g(2/3) = 2
Therefore, f(2/3) = g(2/3), which confirms that our solution x = 2/3 is correct. Verification is an essential step in problem-solving, as it helps to identify any potential errors made during the process. By substituting the solution back into the original equation, we can ensure that the equality holds true. This not only provides confidence in our answer but also reinforces our understanding of the problem and the solution process. In this instance, the verification step clearly demonstrates that x = 2/3 is indeed the value for which f(x) = g(x). This thorough approach to problem-solving is a hallmark of mathematical rigor and is highly recommended in all mathematical endeavors. This complete process ensures accuracy and understanding of the solution.
Conclusion
In conclusion, we have successfully solved for the value of x when f(x) = g(x), given the functions f(x) = -3x + 4 and g(x) = 2. The step-by-step process involved setting up the equation, isolating the variable x, and verifying the solution. We found that x = 2/3 is the value that satisfies the equation. This exercise underscores the importance of algebraic manipulation and function evaluation in solving mathematical problems. The ability to equate functions and solve for variables is a fundamental skill in mathematics, with applications spanning various disciplines. By mastering these techniques, individuals can tackle more complex problems and gain a deeper understanding of mathematical concepts. The methodical approach we followed, from setting up the equation to verifying the solution, is a testament to the importance of precision and accuracy in mathematical problem-solving. Each step was carefully executed, ensuring that the final answer was not only correct but also well-understood. The verification step, in particular, highlighted the significance of checking our work to avoid errors and build confidence in our results. This entire process, from start to finish, exemplifies the beauty and power of mathematics in solving real-world problems and advancing our understanding of the world around us. Therefore, the final answer to the problem is:
x = 2/3
This article serves as a comprehensive guide to solving equations involving function equality, providing a clear and concise methodology for readers to follow. The principles and techniques discussed here can be applied to a wide range of mathematical problems, making this a valuable resource for students, educators, and anyone interested in enhancing their mathematical skills.
