Finding F(-2) For F(x) = 3 * 2^x A Step-by-Step Guide

by Pedro Alvarez 54 views

Hey guys! Today, let's dive into the exciting world of exponential functions and tackle a cool problem. We're given the function f(x) = 3 * 2^x, and our mission, should we choose to accept it (and we totally do!), is to find the value of f(-2). Sounds intriguing, right? Don't worry, it's way simpler than it looks. We'll break it down step by step, so you'll be solving these problems like a pro in no time! This exploration isn't just about crunching numbers; it's about understanding the fundamental nature of exponential functions and how they behave. We'll see how negative exponents play a crucial role and how they transform the way we think about growth and decay. By the end of this, you'll not only know how to find f(-2) but also have a deeper appreciation for the power and elegance of mathematical functions. So, let's buckle up and get ready to embark on this mathematical adventure together! We'll uncover the secrets hidden within the equation and reveal the answer to our quest. Remember, math isn't just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. So, let's unleash our inner mathematicians and conquer this challenge!

Understanding Exponential Functions

Before we jump into solving for f(-2), let's take a moment to understand exponential functions in general. An exponential function is a function where the variable appears in the exponent. The general form looks something like this: f(x) = a * b^x, where a is a constant coefficient, b is the base (a positive number not equal to 1), and x is the exponent. The beauty of exponential functions lies in their ability to model scenarios involving rapid growth or decay. Think about population growth, compound interest, or even the decay of radioactive materials – these phenomena can often be described using exponential functions. Now, let's dissect the key components of our specific function, f(x) = 3 * 2^x. Here, the coefficient a is 3, which acts as a vertical stretch or compression factor. The base b is 2, which dictates the rate of growth. As x increases, the value of 2^x grows exponentially, and multiplying it by 3 simply scales that growth. But what happens when x becomes negative? This is where things get interesting! A negative exponent signifies a reciprocal. In other words, 2^-x is the same as 1 / 2^x. This means that as x becomes more negative, the value of 2^x decreases, approaching zero. This behavior is characteristic of exponential decay. Understanding this fundamental concept of negative exponents is crucial for solving our problem and for grasping the broader applications of exponential functions. So, with this knowledge in our toolkit, we're well-equipped to tackle the challenge of finding f(-2). Let's move on to the next step and see how we can apply this understanding to solve our specific problem.

The Key Role of Negative Exponents

Now, let's zoom in on the key role of negative exponents because they're super important for solving our problem. Remember how we talked about 2^-x being the same as 1 / 2^x? This is the golden rule we need to keep in mind. When we have a negative exponent, it's like we're flipping the base and making the exponent positive. For example, 5^-2 is the same as 1 / 5^2, which equals 1 / 25. This might seem like a small detail, but it's a game-changer when we're dealing with exponential functions. Negative exponents allow us to model situations where things are decreasing or decaying over time. Think about the amount of medicine in your bloodstream as your body metabolizes it, or the value of a car as it depreciates. These scenarios can be elegantly represented using exponential functions with negative exponents. In our case, we're trying to find f(-2) for the function f(x) = 3 * 2^x. This means we need to substitute x with -2, which gives us f(-2) = 3 * 2^-2. See that negative exponent? That's our cue to apply the rule we just learned. We can rewrite 2^-2 as 1 / 2^2. This simple transformation makes the problem much easier to handle. Now we have f(-2) = 3 * (1 / 2^2). We know that 2^2 is 4, so we can further simplify this to f(-2) = 3 * (1 / 4). And finally, multiplying 3 by 1/4 gives us our answer. So, you see, understanding negative exponents is not just about memorizing a rule; it's about unlocking the power of exponential functions to model a wide range of real-world phenomena. With this knowledge under our belts, we're ready to put the final pieces of the puzzle together and find the value of f(-2).

Step-by-Step Solution for f(-2)

Alright, let's get down to business and walk through the step-by-step solution for f(-2). We've laid the groundwork by understanding exponential functions and the crucial role of negative exponents, so now it's time to put that knowledge into action. Remember our function: f(x) = 3 * 2^x. Our mission is to find f(-2), which means we need to substitute x with -2. So, let's do it! We get: f(-2) = 3 * 2^-2 Now, here's where our understanding of negative exponents comes into play. We know that 2^-2 is the same as 1 / 2^2. Let's rewrite our equation: f(-2) = 3 * (1 / 2^2) Next, we need to evaluate 2^2. This is simply 2 multiplied by itself, which equals 4. So, we have: f(-2) = 3 * (1 / 4) Now, we're in the home stretch! We just need to multiply 3 by 1/4. This is the same as dividing 3 by 4, which gives us 3/4. Therefore: f(-2) = 3/4 And that's it! We've successfully found the value of f(-2). It's as simple as substituting the value, applying the rule for negative exponents, and simplifying the expression. Each step builds upon the previous one, leading us to the final answer. This methodical approach is key to solving mathematical problems, especially when dealing with functions. By breaking down the problem into smaller, manageable steps, we can avoid confusion and arrive at the correct solution with confidence. So, give yourself a pat on the back – you've just conquered an exponential function problem! But don't stop here. The more you practice, the more comfortable you'll become with these concepts, and the more complex problems you'll be able to tackle.

Final Answer: f(-2) = 3/4

So, drumroll please… the final answer to our quest is f(-2) = 3/4. Woohoo! We did it! We successfully navigated the world of exponential functions, conquered the negative exponent, and arrived at our destination. It's a pretty satisfying feeling, isn't it? This result tells us that when we input -2 into our function f(x) = 3 * 2^x, the output is 3/4. In other words, the function has a value of 3/4 at the point where x equals -2. But this is more than just a number; it's a point on the graph of the exponential function. If we were to plot this function, the point (-2, 3/4) would lie on the curve. This visual representation can further enhance our understanding of how the function behaves. The exponential function f(x) = 3 * 2^x exhibits exponential decay as x becomes more negative, approaching zero but never actually reaching it. The value of 3/4 at x = -2 is a snapshot of this decay in action. By understanding the interplay between the equation, the graph, and the numerical values, we gain a deeper appreciation for the richness and interconnectedness of mathematics. So, remember, finding the value of a function at a specific point is not just about plugging in a number; it's about understanding the behavior of the function and its place within the broader mathematical landscape. And with that, we've reached the end of our journey. We've not only found the answer but also deepened our understanding of exponential functions. Keep practicing, keep exploring, and keep the mathematical adventures coming!

Practice Problems and Further Exploration

Now that we've cracked the code for finding f(-2), let's keep the momentum going with some practice problems and further exploration. The best way to solidify your understanding of exponential functions is to work through a variety of examples. So, here are a few problems to get you started:

  1. If g(x) = 5 * 3^x, find g(-1).
  2. If h(x) = 2 * (1/2)^x, find h(-3).
  3. If k(x) = 4^-x, find k(2).

Tackling these problems will help you hone your skills in dealing with negative exponents and evaluating exponential functions. Remember to break each problem down step-by-step, applying the principles we've discussed. But the exploration doesn't have to stop there! Exponential functions are fascinating and have numerous applications in the real world. You can delve deeper into the topic by exploring concepts like exponential growth and decay, half-life, and compound interest. These concepts are all closely related to exponential functions and can provide you with a broader perspective on their significance. You can also investigate the graphs of exponential functions and how they change as the base and coefficient vary. Visualizing these functions can provide valuable insights into their behavior. There are tons of resources available online and in textbooks that can help you on your journey. So, don't hesitate to explore, experiment, and ask questions. The world of mathematics is vast and full of exciting discoveries, and exponential functions are just one piece of the puzzle. By continuing to learn and practice, you'll not only master the current topic but also build a strong foundation for future mathematical endeavors. So, go forth and explore the wonders of exponential functions!