Hey everyone! Let's dive into a fun math problem where we need to find the value of x that makes two functions equal to each other. We're given two functions: f(x) = -3x + 4 and g(x) = 2. Our mission, should we choose to accept it (and we do!), is to figure out when f(x) is the same as g(x). This means we want to find the x that makes the equation f(x) = g(x) true. It's like a mathematical treasure hunt, and x is our hidden treasure! To start, let's break down what these functions actually mean and then, step-by-step, we'll solve for x. Think of functions as little machines. You feed them a number (x in this case), and they spit out another number based on a specific rule. For f(x), the rule is: multiply the input (x) by -3 and then add 4. For g(x), the rule is super simple: no matter what you feed it, it always spits out 2. This makes g(x) a constant function because its output never changes. Now, the heart of our problem is setting these two "machines" equal to each other. When we set f(x) = g(x), we're creating an equation that we can solve. This equation will tell us exactly which x value makes these two functions produce the same output. It's like finding the common ground where these two mathematical worlds intersect. So, grab your pencils (or your favorite digital stylus!) and let's embark on this algebraic adventure together. We're about to uncover the x that makes mathematical magic happen!
Setting Up the Equation
Okay, guys, let's get down to brass tacks. Our first crucial step in solving this problem is to actually set up the equation. Remember, we want to find the value of x where f(x) is equal to g(x). We know that f(x) = -3x + 4 and g(x) = 2. So, to set them equal, we simply write: -3x + 4 = 2. Boom! We've got our equation. This equation is the key to unlocking the value of x. It's like a mathematical code that we need to decipher. On the left side of the equation, we have an algebraic expression, -3x + 4, which represents the output of our f(x) function. This expression changes depending on the value of x. The -3x part means we're multiplying x by -3, which will flip the sign of x and make it larger or smaller depending on its value. The + 4 part means we're adding 4 to the result, shifting the whole expression upwards on the number line. On the right side of the equation, we have the constant value 2, which is the output of our g(x) function. It's like a fixed point, a target that our f(x) needs to hit. Now, the equals sign (=) is super important here. It tells us that the expression on the left must have the same value as the number on the right. That's the fundamental rule we need to follow. Think of it like a balancing scale. We need to find the x that makes both sides of the scale perfectly balanced. The next step is to start manipulating this equation, using the rules of algebra, to isolate x and figure out its value. We'll be doing some mathematical maneuvering, but the goal is always to keep the equation balanced. So, with our equation firmly in hand, we're ready to move on to the next phase of our treasure hunt: solving for x!
Isolating x: The Art of Algebraic Manipulation
Alright, buckle up, mathletes! Now comes the fun part: isolating x. This is where we use our algebraic superpowers to get x all by itself on one side of the equation. Our goal is to peel away all the other terms around x until it stands alone in its glorious, solved state. Remember our equation? It's -3x + 4 = 2. The first order of business is to get rid of that + 4 on the left side. To do that, we use the magic of inverse operations. Since we're adding 4, we do the opposite: we subtract 4. But, and this is super important, whatever we do to one side of the equation, we must do to the other side to keep the balance. So, we subtract 4 from both sides:
-3x + 4 - 4 = 2 - 4
This simplifies to:
-3x = -2
Excellent! We've made progress. The + 4 is gone, and we're one step closer to freedom for x. Now, we have -3x = -2. This means -3 multiplied by x equals -2. To get x alone, we need to undo that multiplication. What's the opposite of multiplying by -3? You guessed it: dividing by -3. Again, we need to do this to both sides of the equation to maintain the balance:
(-3x) / -3 = (-2) / -3
This simplifies to:
x = 2/3
Ta-da! We've done it! We've successfully isolated x. It's like we've solved a mathematical puzzle, carefully moving pieces around until the hidden value is revealed. We found that x equals 2/3. But before we declare victory and start the celebration parade, there's one crucial step left: verification. We need to make sure our answer is correct.
Verifying the Solution: The Moment of Truth
Okay, team, this is it – the moment of truth! We've solved for x, and we think it's 2/3. But math, like life, sometimes throws curveballs. So, we need to verify our solution to make sure it actually works. This step is absolutely crucial because it prevents us from confidently marching forward with a wrong answer. Verification is like the safety check on a rocket launch – we want to make sure everything is A-Okay before we blast off. To verify, we take our solution, x = 2/3, and plug it back into the original equation, f(x) = g(x), which we set up as -3x + 4 = 2. We're going to substitute 2/3 for x in the left side of the equation, f(x), and then simplify to see if it equals the right side, g(x), which is 2. So, here we go:
-3 * (2/3) + 4 = ?
First, we multiply -3 by 2/3. Remember, when you multiply a fraction by a whole number, you can think of the whole number as a fraction with a denominator of 1. So, we have:
(-3/1) * (2/3) = -6/3
This simplifies to -2. Now, we plug that back into our equation:
-2 + 4 = ?
-2 + 4 equals 2. So, our equation now looks like this:
2 = 2
Eureka! It checks out! The left side equals the right side. This means that when x = 2/3, f(x) is indeed equal to g(x). Our solution is verified, and we can confidently declare victory. This verification step not only confirms our answer but also deepens our understanding of the problem and the functions involved. We've seen firsthand how the value of x affects the output of f(x) and how it aligns with the constant output of g(x). So, give yourselves a pat on the back, math adventurers! We've successfully navigated this equation and emerged with a verified solution.
Final Answer and Conclusion
Alright, everyone, let's bring this mathematical journey to a close! After all our hard work, we've arrived at the final destination: the solution. We set out to find the value of x for which f(x) = g(x), where f(x) = -3x + 4 and g(x) = 2. We navigated the algebraic terrain, carefully isolating x through a series of strategic moves. We conquered the challenges of inverse operations, maintaining balance and precision every step of the way. And finally, we arrived at our solution: x = 2/3. But we didn't stop there! We knew that true mathematical mastery requires verification. So, we put our solution to the test, plugging it back into the original equation to ensure its validity. And guess what? It passed with flying colors! When x = 2/3, both f(x) and g(x) produce the same output, solidifying our answer. So, with confidence and a touch of mathematical swagger, we can state our final answer: The value of x for which f(x) = g(x) is 2/3. Woohoo! We did it! This problem beautifully illustrates the power of algebra to solve equations and find unknown values. It also highlights the importance of verification in ensuring the accuracy of our solutions. Math isn't just about getting the right answer; it's about understanding the process, the logic, and the relationships between different mathematical concepts. We've learned not only how to solve this specific problem but also valuable skills that can be applied to a wide range of mathematical challenges. So, keep practicing, keep exploring, and keep embracing the joy of mathematical discovery. And remember, even the most complex problems can be conquered with a little bit of patience, a dash of algebraic skill, and a whole lot of determination. Until next time, happy problem-solving!