Hey there, math enthusiasts! Today, we're diving into the fascinating world of functions. Think of a function like a magical machine: you feed it a number (an input), and it spits out another number (an output) based on a specific rule. In this case, our magical machine is the function g(x) = 3(x + 8). The big question is: what happens when we feed this machine the number 12? Let's find out!
Understanding the Function g(x) = 3(x + 8)
Before we jump into calculating g(12), let's break down what this function actually means. The notation "g(x)" simply tells us that we're dealing with a function named "g," and the "x" represents the input we're going to give it. The equation "g(x) = 3(x + 8)" is the heart of the function; it's the rule that dictates how the input "x" is transformed into the output. To understand this better, let's dissect the equation step-by-step.
First, we see "(x + 8)". This means that the first thing our magical machine does is take the input "x" and add 8 to it. So, if we input 5, the machine would calculate 5 + 8 = 13. Simple enough, right? Next, we have "3(x + 8)". The "3" outside the parentheses means we're going to multiply the result of "(x + 8)" by 3. So, continuing with our example, after adding 8 to 5 (getting 13), we would then multiply 13 by 3, resulting in 39. Therefore, g(5) = 39. That's the essence of how this function works: it adds 8 to the input and then multiplies the sum by 3. It's like a little recipe for transforming numbers! Think of functions as a set of instructions. The function g(x) = 3(x + 8) provides a clear, step-by-step guide. First, it tells us to take our input, represented by 'x,' and add 8 to it. This operation is encapsulated within the parentheses, indicating its priority. It's crucial to perform this addition before any other operation, as the order of operations dictates. Once we've added 8 to our input, the next instruction is to multiply the result by 3. This multiplication is indicated by the presence of '3' outside the parentheses, which applies to the entire expression inside the parentheses. This step essentially scales the result of our initial addition, amplifying the effect of adding 8 to the original input. By following these instructions diligently, we can accurately determine the output of the function for any given input. Functions are like mini-programs, each with its own specific set of commands. Understanding these commands is key to unlocking the function's behavior and predicting its output. So, let's keep these instructions in mind as we move forward and apply them to the specific case of g(12). Remember, each step in the function has a purpose, and following them in the correct order is essential for obtaining the correct result. Functions are the building blocks of more complex mathematical models, so mastering their interpretation and application is a valuable skill.
Calculating g(12): Putting the Function to Work
Now that we understand the function g(x) = 3(x + 8), let's tackle the main question: what is the value of g(12)? This simply means we need to replace "x" in the equation with the number 12 and perform the calculations. So, we have g(12) = 3(12 + 8). The first step, as we discussed, is to deal with the parentheses. We need to calculate 12 + 8, which is 20. So now our equation looks like this: g(12) = 3(20). Next, we multiply 3 by 20. This is a straightforward multiplication: 3 * 20 = 60. Therefore, g(12) = 60. That's it! We've successfully calculated the value of the function g(x) when x is 12. The result is 60. This means that when we feed the number 12 into our magical function machine, it spits out the number 60. Let's recap the steps to ensure we fully grasp the process. First, we substituted 'x' with 12 in the function's equation, transforming g(x) = 3(x + 8) into g(12) = 3(12 + 8). Next, we followed the order of operations, prioritizing the expression within the parentheses. We added 12 and 8, resulting in 20. This simplified our equation to g(12) = 3(20). Finally, we performed the multiplication, multiplying 3 by 20 to obtain the result of 60. Thus, we definitively concluded that g(12) equals 60. This calculation showcases the power of functions to transform inputs into outputs based on defined rules. By carefully applying the steps outlined in the function's equation, we can accurately determine the output for any given input. Functions are essential tools in mathematics and computer science, enabling us to model relationships and processes in a precise and predictable manner. Mastering the calculation of function values, like we've done here with g(12), is a fundamental skill for anyone delving into these fields. Remember, the key is to understand the function's equation, substitute the input value correctly, and follow the order of operations meticulously. With practice, these calculations become second nature, empowering you to solve a wide range of mathematical problems. So, let's celebrate our success in finding g(12) and continue exploring the fascinating world of functions!
The Answer and Why It's Important
So, the answer to the question "What is the value of g(12)?" is D. 60. But it's not just about getting the right answer; it's about understanding why the answer is 60. We've seen how the function g(x) transforms inputs, and we've practiced applying this function to a specific value. This process is fundamental to many areas of mathematics and beyond. Functions are the building blocks of more complex mathematical models, and the ability to evaluate them is crucial for solving a wide range of problems. Think about modeling real-world situations: a function could represent the cost of producing a certain number of items, the trajectory of a ball thrown in the air, or even the spread of a disease. Being able to understand and work with functions allows us to make predictions, analyze trends, and ultimately solve problems in a systematic way. It's not just about plugging in numbers; it's about understanding the relationship between variables and the rules that govern them. The ability to evaluate functions extends far beyond the classroom, finding applications in fields such as engineering, economics, computer science, and many more. In engineering, functions are used to model the behavior of circuits, the flow of fluids, and the structural integrity of buildings. Economists use functions to analyze supply and demand, predict market trends, and model economic growth. Computer scientists rely heavily on functions to design algorithms, write software, and manage data. The underlying principles remain the same: functions provide a framework for describing relationships and making predictions. Mastering the art of function evaluation opens doors to a deeper understanding of these fields and empowers individuals to tackle complex challenges with confidence. So, let's remember that the answer to a math problem is not merely a number; it's a gateway to a broader understanding of the world around us. By embracing the concepts behind the calculations, we unlock the true potential of mathematics as a powerful tool for problem-solving and innovation. With each function we evaluate, we strengthen our ability to analyze, predict, and ultimately, shape the world we live in.
Practice Makes Perfect: More Function Fun!
Now that we've conquered g(12), let's keep the momentum going! The best way to truly master functions is to practice. Try plugging in different values for "x" in g(x) = 3(x + 8). What is g(0)? What is g(-8)? What is g(2)? See how the output changes as you change the input. You can even invent your own functions! Maybe try h(x) = 2x - 5, or f(x) = x² + 1. The possibilities are endless! By experimenting with different functions and different inputs, you'll deepen your understanding of how functions work and become a pro at evaluating them. Remember, the key to success in mathematics is consistent effort and a willingness to explore. Functions are like puzzles waiting to be solved, and each solution brings you closer to mastering this essential mathematical concept. So, let's embrace the challenge, dive into the world of functions, and unlock the power of mathematical thinking. We are now armed with the knowledge and skills to tackle any function that comes our way. It's time to put our newfound expertise to the test and continue our journey of mathematical discovery. Remember, practice makes perfect, and the more we engage with functions, the more comfortable and confident we will become in our ability to evaluate them. So, let's not shy away from challenges; let's embrace them as opportunities to learn and grow. With each problem we solve, we strengthen our mathematical foundation and expand our capacity for critical thinking. So, let's celebrate our progress, encourage each other, and continue our exploration of the fascinating world of mathematics. The journey may be challenging at times, but the rewards are well worth the effort. By persevering and embracing the process of learning, we unlock our full potential and pave the way for future success. So, let's keep practicing, keep exploring, and keep having fun with functions!
By understanding the step-by-step process of evaluating functions, we can confidently tackle similar problems and build a solid foundation in mathematics. So, keep practicing, keep exploring, and keep having fun with functions!