Calculating (g O F)(4): A Step-by-Step Guide

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Hey guys! Let's dive into a cool math problem. We're going to figure out the value of (g o f)(4). Don't worry, it's not as scary as it sounds! It's actually a pretty straightforward process, and I'll walk you through it step-by-step. We're given two functions: f(x) and g(x). Our mission? To find the value of the composite function (g o f)(4). This type of problem is super common in algebra, and understanding it is key to building a strong foundation in math. So, let's get started and break it down, shall we?

Understanding the Functions and the Goal

First off, let's clarify what we're working with. We have two functions defined here. These functions are mathematical rules that take an input (x) and produce an output.

  • The function f(x) is defined as: f(x) = (2x - 1) / (x - 3), with the condition that x cannot equal 3. Why the restriction? Because if x were 3, we'd have division by zero, and that's a big no-no in math. This means the domain of f(x) is all real numbers except 3.

  • The function g(x) is defined as: g(x) = 3x + 2. This one is a simpler linear function. It takes an x value, multiplies it by 3, and then adds 2.

So, what are we actually trying to do? We want to find (g o f)(4). The notation (g o f)(4) means that we first apply the function f to the input 4, and then we take the output of f(4) and use it as the input for the function g. Think of it like a two-step process: first, we feed 4 into f, and then we feed the result into g. Our ultimate goal is to find the final value after these two steps. This concept is called function composition. This is like a chain reaction, the output of one function becomes the input of another function. Understanding this sequential process is fundamental to mastering these types of problems. Now, let's do this step by step.

Step-by-step solution

Now, let's take a look at the functions and how we can use them to figure out (g o f)(4). The key to solving this type of problem is to break it down into smaller, manageable steps. This not only makes the process easier to understand but also reduces the chance of making errors. Let's do it!

Step 1: Calculate f(4)

First, we need to find the value of f(4). This means we substitute x with 4 in the function f(x). The formula is f(x) = (2x - 1) / (x - 3). So, let's plug in the number 4: f(4) = (2 * 4 - 1) / (4 - 3). We perform the calculations within the parentheses first:

f(4) = (8 - 1) / (1) f(4) = 7 / 1 f(4) = 7

So, f(4) = 7. This is the first part of our journey. This result is the output of the function f(x) when the input is 4.

Step 2: Calculate g(f(4))

Now that we know f(4) = 7, we can find g(f(4)). Since f(4) = 7, we are essentially looking for g(7). We use the function g(x) = 3x + 2 and substitute x with 7: g(7) = (3 * 7) + 2. Now let's calculate this:

g(7) = 21 + 2 g(7) = 23

So, g(7) = 23. This is the final step. We started with the input of x = 4, we put it in the function f(x), and then the result of f(x), we put it into the function g(x), and our answer is 23.

Therefore, the value of (g o f)(4) is 23. This approach clearly demonstrates how the composition of functions works.

Conclusion and Answer Choice

We've successfully computed the value of (g o f)(4) which is 23. So, the correct answer among the options provided is D. 23. This problem perfectly illustrates the concept of function composition and how to evaluate composite functions step by step. By systematically substituting the values and following the order of operations, we reached our final answer. Remember, practice makes perfect. The more you work through these types of problems, the more comfortable and confident you'll become! So, keep up the great work, and you'll be acing these questions in no time!

Why Understanding Function Composition Matters

Understanding function composition isn't just about solving a math problem; it's about grasping a fundamental concept that appears in many areas of mathematics and computer science. From calculus to advanced programming, the idea of applying one function after another is a recurring theme.

  • In Calculus: Function composition is essential when dealing with the chain rule for differentiation. The chain rule helps you find the derivative of a composite function, which is a core concept in understanding how functions change.

  • In Computer Science: The concept of function composition is used in functional programming. The idea of combining functions to create new, more complex functions is very powerful. This type of programming allows you to write efficient and concise code by reusing smaller, modular functions.

  • In Real-World Applications: Function composition can model real-world processes where one action leads to another. Think of a manufacturing process where raw materials are processed through multiple stages, or a financial model where investments are subject to various interest rates and fees. Each step can be seen as a function, and the overall process is a composition of those functions.

So, grasping this concept of function composition really sets you up for success in further mathematical and computing challenges. Keep practicing, keep learning, and you will do great!