Finding (h∘f)(3): A Step-by-Step Guide

Have you ever wondered how to combine two functions? In mathematics, this is known as a composite function. Today, we're going to explore a specific example of a composite function. We'll break down how to find (h∘f)(3) when given two functions: f(x) = 4x - 1 and h(x) = -x + 1. It might sound intimidating at first, but don't worry, we will explain every step in detail, making it easy to understand and apply. Understanding composite functions is a fundamental skill in algebra and calculus, so let's dive in!

Understanding Composite Functions

Before we jump into the calculation, let's make sure we're all on the same page about what a composite function actually is. Composite functions might sound complex, but they're simply a way of combining two functions by applying one function to the result of another. Think of it like a mathematical assembly line: you have an input, it goes through one function, and then the output of that function becomes the input for the second function.

The notation (h∘f)(x) represents the composite function where we first apply the function f to x, and then we apply the function h to the result. In other words, (h∘f)(x) = h(f(x)). The small circle "∘" signifies composition. It's crucial to understand this notation because the order matters! (h∘f)(x) is generally not the same as (f∘h)(x). Imagine you have two machines: one that doubles a number and another that adds 3. If you put a number through the doubling machine first and then the adding machine, you'll get a different result than if you put it through the adding machine first and then the doubling machine.

Now, let's break down why composite functions are so important. They allow us to model complex relationships by breaking them down into simpler steps. For instance, in physics, you might use a composite function to describe the motion of an object under the influence of multiple forces. In computer science, composite functions can be used to build complex algorithms from simpler subroutines. And in everyday life, you might unknowingly use the concept of composite functions when following a recipe (first you chop the vegetables, then you sauté them, and so on). Grasping the concept of function composition opens doors to understanding more advanced mathematical concepts and real-world applications. So, with this fundamental understanding in place, we're ready to tackle our specific problem.

Step 1: Evaluate f(3)

The first step in finding (h∘f)(3) is to evaluate f(3). This means we need to substitute x = 3 into the expression for f(x). Remember, we're given that f(x) = 4x - 1. So, wherever we see an 'x' in the function definition, we replace it with the number 3.

Let's do the substitution: f(3) = 4(3) - 1. Now, we just need to simplify this expression. Following the order of operations (PEMDAS/BODMAS), we first perform the multiplication: 4 multiplied by 3 equals 12. So, our expression becomes f(3) = 12 - 1. Finally, we perform the subtraction: 12 minus 1 equals 11. Therefore, f(3) = 11. This result is crucial because it becomes the input for our next step. We've essentially completed the first part of our mathematical assembly line. Calculating f(3) correctly is essential because it forms the foundation for the rest of the problem. Any error in this step will propagate through the subsequent calculations, leading to an incorrect final answer.

So, to recap, we've taken the input value of 3 and applied the function f to it, resulting in an output value of 11. This output now becomes the input for the next function in our composite function.

Step 2: Evaluate h(f(3))

Now that we know f(3) = 11, we can move on to the next step: evaluating h(f(3)). Remember, (h∘f)(3) is the same as h(f(3)), which means we need to find the value of the function h when its input is f(3). Since we've already determined that f(3) = 11, we can rewrite this as h(11). This step beautifully illustrates the core concept of composite functions: the output of one function becomes the input of another.

We are given that h(x) = -x + 1. Just like in the previous step, we substitute the input value (in this case, 11) for 'x' in the function's expression. So, h(11) = -(11) + 1. Notice the importance of the negative sign in front of the x in the function h(x). It's a common source of errors if overlooked. Now, we simplify the expression. -11 + 1 equals -10. Therefore, h(11) = -10.

This means that when we input the value 11 into the function h, the output is -10. Since 11 was the output of f(3), we've essentially traced the entire composite function: 3 goes into f, f outputs 11, 11 goes into h, and h outputs -10. Evaluating h(f(3)) accurately is the final piece of the puzzle. It combines our understanding of function evaluation and composite functions to arrive at the solution. We have successfully applied both functions in the correct order.

Step 3: State the Result

We've done all the hard work, and now it's time to state our result clearly. We've found that (h∘f)(3) = h(f(3)) = h(11) = -10. Therefore, the final answer is -10. It's important to present the result in a clear and concise manner, showing the logical flow of the steps we took. This not only ensures that the answer is easily understood but also reinforces our own understanding of the process.

Clearly stating the result is crucial for effective communication in mathematics. It's not just about getting the right answer; it's also about being able to explain your reasoning and justify your solution. So, whenever you solve a mathematical problem, make sure to conclude with a clear statement of your final answer.

Conclusion

In this step-by-step guide, we've successfully found (h∘f)(3) given f(x) = 4x - 1 and h(x) = -x + 1. We've broken down the process into manageable steps: first, we evaluated f(3), then we used that result to evaluate h(f(3)), and finally, we stated our answer. This example demonstrates the fundamental concept of composite functions: applying one function to the result of another. Mastering composite functions is a key skill in mathematics, opening the door to more advanced topics in algebra, calculus, and beyond.

Remember, the order of operations is crucial when dealing with composite functions. (h∘f)(x) is generally different from (f∘h)(x). Always work from the inside out: evaluate the innermost function first, and then use its output as the input for the next function. Practice makes perfect, so try working through similar examples with different functions and input values. The more you practice, the more comfortable you'll become with the concept of composite functions.

For further exploration and to solidify your understanding, you might find helpful resources on websites like Khan Academy's Composite Functions Section. This will provide you with more examples and exercises to hone your skills. Keep practicing, and you'll become a pro at working with composite functions!