Solving F(g(2)) With F(x) = 2x^2 + 1 And G(x) = 3x - 5

Hey everyone! Today, let's dive into a fun little math problem where we'll be evaluating composite functions. Specifically, we're tasked with finding the value of f(g(2)), given that f(x) = 2x^2 + 1 and g(x) = 3x - 5. This might sound a bit intimidating at first, but trust me, it's a pretty straightforward process once you break it down. We will provide a detailed step-by-step solution and explanation to clarify how to solve this type of problem, and hopefully, by the end of this article, you'll feel confident tackling similar problems on your own.

Understanding Composite Functions

Before we jump into the calculation, it's important to have a solid understanding of what composite functions are and how they work. Think of it like this: a composite function is essentially a function within a function. In our case, f(g(x)) means we're taking the output of the function g(x) and plugging it into the function f(x). The notation f(g(2)) specifically means that we need to first evaluate g(x) at x = 2, and then take that result and plug it into f(x). Breaking down complex problems into smaller, more manageable steps is a key strategy in mathematics, and composite functions are a perfect example of this. Many real-world scenarios can be modeled using composite functions, such as calculating the cost of an item after a discount and sales tax, or modeling the population growth of a species over time, where one function depends on the output of another function. By understanding the fundamental concept of plugging the output of one function into another, you'll be well-equipped to tackle more complex mathematical models and applications.

Composite functions are a cornerstone of mathematical analysis, especially in calculus, where they are fundamental in understanding the chain rule, a crucial technique for differentiating complex functions. The chain rule allows us to find the derivative of a composite function by breaking it down into the derivatives of its constituent functions, making it easier to analyze the rate of change of the composite function. For instance, if you are trying to optimize a process or system that involves multiple steps, understanding how changes in the input variables propagate through the system requires a firm grasp of composite functions and their derivatives. In computer science, composite functions are used to design complex algorithms by layering simpler functions, creating modular and reusable code. Each layer performs a specific task, and the output of one layer becomes the input for the next, forming a sophisticated processing pipeline. This approach enhances the maintainability and scalability of software systems, as individual components can be modified or replaced without affecting the entire system. So, understanding composite functions extends beyond basic algebra and calculus; it's a critical skill for various fields that rely on mathematical modeling and computational thinking.

Step 1: Evaluate g(2)

The first step in finding f(g(2)) is to evaluate g(2). Remember, we're given that g(x) = 3x - 5. So, to find g(2), we simply substitute x with 2 in the expression for g(x). This gives us g(2) = 3(2) - 5. Now, let's do the math. First, we multiply 3 by 2, which gives us 6. Then, we subtract 5 from 6, which leaves us with 1. So, we've found that g(2) = 1. This first step is crucial because it provides the input for the next step, which is evaluating f(x). Think of it like a machine where g(x) is the first stage, processing the input x and producing an output. This output then becomes the input for the next stage, f(x). If we were to skip this step or calculate it incorrectly, the final result would also be incorrect. Therefore, paying close attention to detail and ensuring accuracy in each step is essential when dealing with composite functions. Mastering the evaluation of g(2) sets the stage for a seamless transition to the next step, where we'll use this result to find f(g(2)). Remember, math often builds upon itself, so a solid foundation in the initial steps makes the subsequent steps much easier to handle.

The substitution method used here is a fundamental concept in algebra and is applied extensively in various mathematical contexts. In addition to evaluating functions, it is used in solving equations, simplifying expressions, and even in more advanced topics like calculus. The key is to replace a variable with its given value, ensuring that the operation is performed correctly according to the order of operations (PEMDAS/BODMAS). For instance, in physics, when calculating the potential energy of an object, one might substitute the height and mass of the object into the potential energy formula, PE = mgh. Similarly, in economics, when determining the equilibrium price and quantity in a market, one might substitute the demand and supply functions to find the point where they intersect. In computer programming, function calls work similarly, where input values (arguments) are substituted into the function's parameters, and the function executes its code based on these values. The consistent application of substitution across these domains highlights its versatility and significance. Understanding and practicing this basic skill can greatly improve your problem-solving abilities in mathematics and other quantitative fields. So, always take the time to ensure your substitutions are accurate and your calculations are precise, as this lays the groundwork for more complex operations.

Step 2: Evaluate f(g(2)) which is f(1)

Now that we know g(2) = 1, we can move on to the second step: evaluating f(g(2)), which is the same as evaluating f(1). Remember, we're given that f(x) = 2x^2 + 1. So, to find f(1), we substitute x with 1 in the expression for f(x). This gives us f(1) = 2(1)^2 + 1. Now, let's break down the calculation. First, we need to square 1, which is simply 1 * 1 = 1. Then, we multiply that result by 2, which gives us 2 * 1 = 2. Finally, we add 1 to 2, which results in 3. Therefore, f(1) = 3. This means that f(g(2)) = 3. This final step demonstrates the power of composite functions: we've taken an input (2), transformed it using one function (g(x)), and then transformed the result again using another function (f(x)) to arrive at our final output (3). Each step is crucial, and the order matters. If we tried to evaluate f(x) first and then g(x), we would get a completely different answer.

Evaluating f(1) involves adhering to the order of operations, a fundamental principle in mathematics that ensures consistent and accurate calculations. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serves as a guide for the sequence in which operations should be performed. In our case, we first dealt with the exponent (1^2), then performed the multiplication (2 * 1), and finally completed the addition (2 + 1). Following the order of operations is not just a mathematical convention but a logical necessity. Ignoring this order can lead to drastically different results, which is why it's emphasized in elementary mathematics and remains relevant in more advanced topics. For example, in computer programming, compilers and interpreters rely on the order of operations to correctly execute arithmetic expressions. A mistake in the sequence can cause a program to crash or produce incorrect results, highlighting the importance of precision. In fields like engineering and physics, where calculations are crucial for design and analysis, adhering to the order of operations ensures that models and predictions are accurate and reliable. Therefore, consistently applying this principle is key to ensuring the integrity and correctness of mathematical and computational work.

Conclusion

So, to recap, we've successfully found that f(g(2)) = 3 when f(x) = 2x^2 + 1 and g(x) = 3x - 5. We did this by first evaluating g(2) and then using that result to evaluate f(x). The key takeaway here is understanding how composite functions work – that is, how the output of one function becomes the input of another. By breaking down the problem into smaller, manageable steps and carefully performing each calculation, we arrived at the correct answer. I hope this explanation has been helpful and has boosted your confidence in tackling similar problems. Remember, practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! If you have any questions or want to explore more examples, feel free to ask. Keep exploring the world of mathematics, guys, and you'll discover all sorts of cool and interesting things!

Understanding composite functions opens the door to more advanced mathematical concepts and applications, such as calculus, differential equations, and mathematical modeling. In calculus, the chain rule, as mentioned earlier, is a direct application of the composite function concept, allowing us to differentiate functions within functions. In differential equations, composite functions can appear in solutions and transformations, enabling the analysis of complex systems and behaviors. Mathematical modeling often involves creating functions that represent real-world phenomena, and these functions can be combined to represent more intricate interactions. For example, in epidemiology, the spread of a disease can be modeled using composite functions that account for factors such as infection rates, recovery rates, and population density. In economics, composite functions can be used to model the impact of government policies on economic indicators like GDP and unemployment. In engineering, complex systems like control systems and signal processing circuits rely heavily on composite functions to analyze and design their behavior. The ability to work with composite functions is therefore not just a mathematical skill but a valuable asset in numerous fields, enabling a deeper understanding and manipulation of complex systems and processes. By mastering this fundamental concept, you're not just learning math; you're gaining a powerful tool for problem-solving and analysis in a wide range of domains.