Solving Function Composition Problems Step-by-Step

Hey guys! Let's dive into this math problem together. We've got two functions here, and we need to figure out how they work when they're combined. It might seem a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. We will explore the concept of function composition, a fundamental operation in mathematics, especially in algebra and calculus. Function composition essentially involves applying one function to the result of another. In simpler terms, it's like a chain reaction where the output of one function becomes the input of the next. This operation allows us to create more complex functions from simpler ones, providing a powerful tool for modeling and solving various problems. In this guide, we will walk through a specific problem involving function composition, highlighting the key steps and principles involved. By understanding the mechanics of function composition, you'll gain a valuable skill for tackling a wide range of mathematical challenges. So, grab your thinking caps, and let's get started on unraveling the intricacies of function composition!

Understanding the Problem

Okay, so the problem gives us two functions: t(x) = 2x - 4 and (g o t)(x) = 4x² - 24x + 32. The notation (g o t)(x) might look a bit strange, but it just means we're plugging the function t(x) into the function g(x). Think of it like this: t(x) is doing its thing first, and then g(x) takes the result and does its thing with it. The main goal here is to figure out what the function g(x) actually is. We're not given g(x) directly, but we're given enough information to work it out. Function composition is a crucial concept in mathematics, allowing us to build more complex functions from simpler ones. It's like creating a machine within a machine, where the output of the inner machine becomes the input for the outer one. Understanding function composition is essential for solving a variety of problems in algebra, calculus, and other areas of mathematics.

Let's break down the notation (g o t)(x) further. The small circle "o" represents the composition operation. It tells us to apply the function on the right (t(x) in this case) first, and then apply the function on the left (g(x)) to the result. So, (g o t)(x) is the same as g(t(x)). This means we first evaluate t(x), and then we take that value and plug it into g(x). Visualizing this process can help to grasp the concept more easily. Imagine t(x) as a machine that takes x as input and produces a certain output. Then, g(x) is another machine that takes the output of t(x) as its input and produces a final output. Our task is to figure out the inner workings of the g(x) machine.

Now, let's think about how we can use the information we have to find g(x). We know t(x) = 2x - 4, and we know (g o t)(x) = 4x² - 24x + 32. Since (g o t)(x) is the same as g(t(x)), we can substitute t(x) into g(x). This means we're looking for a function g such that when we plug in 2x - 4 (which is t(x)), we get 4x² - 24x + 32. This might seem like a bit of a puzzle, but we can solve it using algebraic techniques. The key is to recognize that the expression 4x² - 24x + 32 is the result of applying g(x) to 2x - 4. So, we need to work backwards to figure out what g(x) must be. This involves some algebraic manipulation and pattern recognition, but with a systematic approach, we can crack this problem. The process of finding g(x) given (g o t)(x) and t(x) is a common type of problem in function composition, and mastering this technique will be very helpful in your mathematical journey.

Finding g(x)

Alright, so to find g(x), we're going to use a little trick called substitution. We know that t(x) = 2x - 4. Let's make things simpler by saying y = 2x - 4. This means we can rewrite x in terms of y. If we solve y = 2x - 4 for x, we get x = (y + 4) / 2. Now we can substitute this expression for x into (g o t)(x). Remember, (g o t)(x) = 4x² - 24x + 32. So, we're going to replace every x in that expression with (y + 4) / 2. Substitution is a powerful technique in algebra that allows us to simplify complex expressions by replacing one variable with another. In this case, we are using substitution to express (g o t)(x) in terms of y, which will help us isolate and determine the function g(x). The key idea is to find a suitable substitution that simplifies the given equation and allows us to solve for the unknown function. By carefully choosing our substitution, we can transform the problem into a more manageable form.

Let's go through the substitution step by step. We have (g o t)(x) = 4x² - 24x + 32. We also have x = (y + 4) / 2. So, we replace each x in the expression for (g o t)(x) with (y + 4) / 2. This gives us:

4 * [((y + 4) / 2)²] - 24 * [(y + 4) / 2] + 32

Now we need to simplify this expression. First, let's simplify the square term: [((y + 4) / 2)²] = (y² + 8y + 16) / 4. Then, we multiply this by 4: 4 * [(y² + 8y + 16) / 4] = y² + 8y + 16. Next, let's simplify the second term: 24 * [(y + 4) / 2] = 12 * (y + 4) = 12y + 48. Now we have:

y² + 8y + 16 - (12y + 48) + 32

Let's distribute the negative sign and combine like terms: y² + 8y + 16 - 12y - 48 + 32. Combining the y terms gives us 8y - 12y = -4y. Combining the constant terms gives us 16 - 48 + 32 = 0. So, our simplified expression is:

y² - 4y

This means that g(y) = y² - 4y. Remember, we substituted y = 2x - 4. Now that we've found g(y), we can simply replace y with x to get g(x). So, g(x) = x² - 4x. This is our answer for the function g(x). We have successfully found g(x) by using the substitution method and simplifying the resulting expression. This process demonstrates the power of algebraic manipulation in solving function composition problems.

Finding g(x-1)

Okay, we've found g(x), which is x² - 4x. Now the question asks us to find g(x - 1). This is another application of function evaluation. To find g(x - 1), we simply replace every x in the expression for g(x) with (x - 1). So, we're going to plug in (x - 1) into x² - 4x. This gives us: (x - 1)² - 4(x - 1). Function evaluation is a fundamental skill in mathematics. It involves substituting a given value or expression into a function to determine the output. In this case, we are evaluating the function g(x) at the point (x - 1). This means we are replacing every instance of x in the function's formula with the expression (x - 1). Understanding function evaluation is crucial for working with functions and solving various mathematical problems.

Let's break down the process step by step. We have g(x - 1) = (x - 1)² - 4(x - 1). First, let's expand the square term: (x - 1)² = (x - 1)(x - 1) = x² - 2x + 1. Then, let's distribute the -4 in the second term: -4(x - 1) = -4x + 4. Now we have:

g(x - 1) = x² - 2x + 1 - 4x + 4

Next, we combine like terms. The x² term stays as it is. Combining the x terms gives us -2x - 4x = -6x. Combining the constant terms gives us 1 + 4 = 5. So, our simplified expression for g(x - 1) is:

g(x - 1) = x² - 6x + 5

This is our final answer for g(x - 1). We have successfully found g(x - 1) by substituting (x - 1) into the expression for g(x) and simplifying the resulting expression. This process demonstrates the importance of careful algebraic manipulation and combining like terms to arrive at the correct answer. Evaluating functions at different points or expressions is a common task in mathematics, and mastering this skill is essential for success in various mathematical contexts.

Putting it All Together

So, to recap, we started with the functions t(x) = 2x - 4 and (g o t)(x) = 4x² - 24x + 32. We used substitution to find that g(x) = x² - 4x. Then, we substituted (x - 1) into g(x) to find g(x - 1) = x² - 6x + 5. That's it! We've solved the problem step by step. This problem demonstrates the power of breaking down complex mathematical problems into smaller, more manageable steps. By carefully applying algebraic techniques and understanding the concept of function composition, we were able to successfully determine the function g(x) and evaluate it at a specific point. Remember, practice makes perfect! The more you work with function composition and function evaluation, the more comfortable and confident you will become in solving these types of problems.

Function composition is a fundamental concept in mathematics that has wide-ranging applications in various fields, including physics, engineering, and computer science. Understanding function composition allows us to model complex systems and solve intricate problems by breaking them down into smaller, more manageable components. The ability to compose functions and evaluate them at different points is a valuable skill that will serve you well in your mathematical journey and beyond. Keep practicing and exploring the world of functions, and you will discover the beauty and power of mathematics. Don't be afraid to ask questions and seek help when needed. Learning mathematics is a collaborative process, and we are all in this together. Keep up the great work, and you will achieve your mathematical goals!

So there you have it, guys! We've tackled a tricky function composition problem and come out on top. Remember, the key is to break things down, use substitution wisely, and take it one step at a time. Keep practicing these kinds of problems, and you'll become a function composition pro in no time! Function composition is a powerful tool in mathematics that allows us to combine functions in interesting ways. By mastering the techniques discussed in this guide, you will be well-equipped to tackle a wide range of mathematical challenges. Remember, mathematics is not just about memorizing formulas, it's about understanding concepts and developing problem-solving skills.

The problem we solved today involved finding the function g(x) given (g o t)(x) and t(x), and then evaluating g(x) at (x - 1). This type of problem is common in algebra and calculus, and it tests your understanding of function composition and function evaluation. The steps we followed included substitution, simplification, and careful algebraic manipulation. By breaking the problem down into smaller steps, we were able to solve it systematically and efficiently. The key takeaway from this problem is the importance of understanding the notation and the definitions of function composition and function evaluation. Once you have a solid grasp of these concepts, you can apply the techniques we discussed to solve a variety of similar problems.

Finally, remember that mathematics is a journey, not a destination. There will be challenges along the way, but with perseverance and practice, you can overcome them and achieve your mathematical goals. Don't be afraid to make mistakes, because mistakes are opportunities to learn and grow. Keep exploring the world of mathematics, and you will discover its beauty and power. Function composition is just one piece of the puzzle, but it's an important piece that will help you build a strong foundation in mathematics. So, keep practicing, keep learning, and keep exploring! You got this!