Solving For X When (f-g)(x) Equals Zero

Hey there, math enthusiasts! Ever wondered where two functions might just cross paths or, in mathematical terms, where their difference equals zero? Today, we're diving deep into a fascinating problem that involves finding exactly that point. We've got two functions, f(x) = 16x - 30 and g(x) = 14x - 6, and our mission, should we choose to accept it, is to pinpoint the value of x that makes (f - g)(x) = 0. Buckle up, because we're about to embark on a mathematical journey filled with twists, turns, and ultimately, a satisfying solution. Let's break down the problem, explore the concepts, and emerge victorious with a clear understanding of how to tackle such questions.

Unpacking the Functions: f(x) and g(x)

Before we jump into the nitty-gritty, let's take a moment to appreciate the stars of our show: the functions f(x) and g(x). These aren't just any equations; they're mathematical machines that take an input (x) and, after a bit of processing, spit out an output. Think of f(x) = 16x - 30 as a machine that multiplies your input by 16 and then subtracts 30. Similarly, g(x) = 14x - 6 multiplies your input by 14 and subtracts 6. These functions are linear, meaning they represent straight lines when graphed, and their behavior is predictable and elegant. Understanding their individual personalities is crucial before we can analyze their relationship, especially when we're looking for the point where their difference vanishes. We need to grasp what each function does independently before we can understand what happens when we put them together in the expression (f - g)(x). So, let’s keep these functions in mind as we proceed, because they’re the key to unlocking the solution.

Deciphering (f - g)(x): What Does It Mean?

Now that we're acquainted with our functions, let's decode the expression (f - g)(x). This notation might seem a bit cryptic at first, but it's actually quite straightforward. It simply means subtracting the function g(x) from the function f(x). In other words, for any given value of x, we first evaluate f(x), then evaluate g(x), and finally subtract the latter from the former. This resulting value represents the difference between the outputs of the two functions at that particular x. Geometrically, (f - g)(x) can be visualized as the vertical distance between the graphs of f(x) and g(x) at a specific point. When we set (f - g)(x) = 0, we're essentially asking: at what x value do these two functions produce the same output, or where do their graphs intersect? This concept is pivotal in understanding the problem, as it transforms the question into a search for the intersection point of two lines. Grasping this interpretation is crucial for not only solving this problem but also for understanding the broader applications of function subtraction in mathematics and beyond. So, let’s keep this key understanding in mind as we move forward.

The Quest for Zero: Setting (f - g)(x) = 0

Our main objective, as you might recall, is to find the value of x for which (f - g)(x) = 0. This equation is the heart of our problem, and solving it will lead us to the solution. To tackle this, we first need to explicitly determine the expression for (f - g)(x). Remember, this means subtracting the entire function g(x) from the entire function f(x). This step is crucial because it simplifies the problem into a manageable algebraic equation. Once we have the expression for (f - g)(x), we can set it equal to zero and solve for x. This process involves using basic algebraic techniques such as combining like terms and isolating the variable. It's like a mathematical treasure hunt, where each step brings us closer to the coveted value of x. The beauty of this approach lies in its directness and clarity, transforming a seemingly complex problem into a series of straightforward steps. So, let's roll up our sleeves and get ready to apply our algebraic skills to find the value of x that makes (f - g)(x) equal to zero.

Crunching the Numbers: Solving the Equation

Alright, let's get down to business and crunch some numbers! We know that f(x) = 16x - 30 and g(x) = 14x - 6. To find (f - g)(x), we subtract g(x) from f(x):

(f - g)(x) = (16x - 30) - (14x - 6)

Now, let's simplify this expression. Remember to distribute the negative sign carefully:

(f - g)(x) = 16x - 30 - 14x + 6

Next, we combine like terms:

(f - g)(x) = (16x - 14x) + (-30 + 6)

(f - g)(x) = 2x - 24

Great! Now we have a simplified expression for (f - g)(x). Our next step is to set this expression equal to zero and solve for x:

2x - 24 = 0

To isolate x, we first add 24 to both sides of the equation:

2x = 24

Finally, we divide both sides by 2:

x = 12

Eureka! We've found the value of x that makes (f - g)(x) = 0. It's a testament to the power of careful algebra and step-by-step problem-solving. So, let's take a moment to appreciate our hard work and the beautiful solution we've uncovered.

The Grand Finale: The Value of x

After our mathematical expedition, we've arrived at the final destination: the value of x. Through careful subtraction, simplification, and equation-solving, we've determined that (f - g)(x) = 0 when x = 12. This single number is the answer to our initial question, the sweet spot where the difference between the two functions vanishes. But it's more than just a number; it's a symbol of our problem-solving journey, a testament to the power of mathematical reasoning. This value tells us that at x = 12, the outputs of f(x) and g(x) are identical, marking the point where their graphs intersect. This understanding not only answers the specific question but also provides a deeper insight into the relationship between the two functions. So, let's celebrate our victory and the elegance of mathematics that has guided us to this satisfying conclusion. This journey reminds us that every problem, no matter how daunting it may seem, can be conquered with a methodical approach and a dash of mathematical curiosity.

Reflecting on the Journey: What Did We Learn?

As we reach the end of our mathematical adventure, let's pause for a moment and reflect on the journey. We didn't just find an answer; we explored concepts, honed our skills, and gained a deeper understanding of functions and their relationships. We started with two functions, f(x) and g(x), and a question: for what value of x does their difference equal zero? Along the way, we learned how to interpret the notation (f - g)(x), how to subtract functions, and how to solve linear equations. We saw how a seemingly complex problem can be broken down into smaller, manageable steps. More importantly, we experienced the satisfaction of solving a problem through logical reasoning and mathematical techniques. This journey underscores the power of a systematic approach and the importance of understanding the underlying concepts. It's a reminder that mathematics is not just about numbers and equations; it's about problem-solving, critical thinking, and the joy of discovery. So, let's carry these lessons forward, ready to tackle new mathematical challenges with confidence and enthusiasm.

Extra Practice: Test Your Skills!

Now that you've mastered the art of finding where (f - g)(x) = 0, it's time to put your skills to the test! Here are a few practice problems to help solidify your understanding:

1. If f(x) = 5x + 7 and g(x) = 2x - 1, for which value of x does (f - g)(x) = 0?
2. Given f(x) = -3x + 10 and g(x) = -5x + 4, find the x value that satisfies (f - g)(x) = 0.
3. Let f(x) = x/2 + 3 and g(x) = x - 1. Determine the value of x for which (f - g)(x) = 0.

These problems offer a variety of scenarios to challenge your abilities. Remember to follow the same steps we used in the original problem: first, find the expression for (f - g)(x), then set it equal to zero, and finally, solve for x. Don't be afraid to make mistakes; they're a natural part of the learning process. The key is to persevere, apply what you've learned, and enjoy the process of mathematical exploration. Happy solving!

Wrapping Up: The Beauty of Functions

So, there you have it, folks! We've successfully navigated the world of functions, tackled a challenging problem, and emerged with a clear solution and a deeper understanding. We've seen how functions can be combined, how their differences can be analyzed, and how equations can be solved to reveal hidden relationships. This journey highlights the beauty and power of mathematics, its ability to describe and explain the world around us. Functions, in particular, are fundamental building blocks in mathematics and have applications in countless fields, from physics and engineering to economics and computer science. By mastering the concepts we've explored today, you're not just solving problems; you're building a foundation for future mathematical endeavors. So, keep exploring, keep questioning, and keep embracing the beauty of functions and the magic of mathematics. Until next time, happy calculating!