Evaluate (g∘h)(4): Step-by-Step Function Composition

by Viktoria Ivanova 53 views

Hey everyone! Today, we're diving into the world of composite functions. Specifically, we're going to tackle a problem that involves evaluating a composite function for a given value of x. Don't worry if that sounds intimidating – we'll break it down step by step so it's super easy to understand. We will solve the problem where we have two functions, g(x) and h(x), and we need to find (g ∘ h)(4). This notation might look a bit strange if you're new to composite functions, but it's simply asking us to plug h(4) into the function g(x). Think of it like a chain reaction: first, we evaluate h(4), and then we take that result and plug it into g(x). This process is what makes composite functions so interesting and powerful in mathematics. They allow us to combine different functions and create new relationships between variables. Before diving into the solution, let’s have a clearer picture of what composite functions exactly are. Composite functions are created when one function is plugged into another. In mathematical terms, a composite function is a function that is written inside another function. So, let’s get started and solve this problem together! By the end of this guide, you’ll be a pro at evaluating composite functions.

Understanding Composite Functions

Before we jump into the problem, let's make sure we're all on the same page about composite functions. Imagine you have two machines. The first machine, h(x), takes an input, does some work on it, and spits out an output. Then, the second machine, g(x), takes that output as its own input, does its thing, and produces a final result. That's essentially what a composite function is! This process is also called function composition. Function composition is a fundamental operation in mathematics where the output of one function becomes the input of another function. This concept is crucial in various mathematical fields, including calculus, algebra, and analysis. Composite functions help us understand how different functions interact and how their combined behavior can be analyzed. In real-world applications, composite functions can model complex systems where multiple processes are interconnected. For example, in economics, composite functions can represent how changes in production costs affect the final price of a product, taking into account multiple stages of manufacturing and distribution. In computer science, composite functions are used in the design of algorithms and data structures, where operations are sequentially applied to data. Mathematically, we write a composite function as (g ∘ h)(x), which is read as "g of h of x". This means we first apply the function h to x, and then we apply the function g to the result. This order is very important, as (g ∘ h)(x) is generally not the same as (h ∘ g)(x). The notation (g ∘ h)(x) emphasizes that the function h is applied first, and its output is then used as the input for the function g. This sequential application is the core concept behind function composition, and understanding this order is critical for correctly evaluating and interpreting composite functions. Now that we've got a solid grasp of the basics, let's move on to the specific problem we're tackling today. Remember, the key is to take it one step at a time and focus on what each function is doing.

Problem Setup: Defining the Functions

Okay, let's get down to business. Our problem gives us two functions:

  • g(x) = √(5x)
  • h(x) = x³ + 6x

And we need to find (g ∘ h)(4). Remember what we discussed earlier? This means we need to find g(h(4)). So, the first step is to figure out what h(4) actually is. When we are presented with functions like these, it’s crucial to understand their individual behaviors. The function g(x) = √(5x) involves a square root, which means the input to the square root (in this case, 5x) must be non-negative. This implies that the domain of g(x) includes only values of x that are greater than or equal to zero. The square root function itself is a fundamental concept in mathematics, used extensively in geometry, physics, and engineering. It represents the inverse operation of squaring, and understanding its properties is essential for solving equations and modeling real-world phenomena. For instance, in physics, square roots appear in calculations related to velocity, energy, and distance. Similarly, the function h(x) = x³ + 6x is a polynomial function. Polynomial functions are known for their smooth and continuous behavior, making them widely used in mathematical modeling. The term x³ indicates a cubic function, which has a distinctive shape with possible local maxima and minima. The additional term 6x adds a linear component, affecting the overall shape and behavior of the function. Polynomial functions are used in various applications, such as curve fitting, interpolation, and optimization problems. In calculus, they are particularly important because they are easy to differentiate and integrate. The combination of these two functions in the composite function (g ∘ h)(x) creates a more complex relationship, requiring us to understand how the output of h(x) becomes the input of g(x). This interplay between different types of functions is a key aspect of composite functions and their applications in more advanced mathematics. Before we can evaluate the composite function, we need to compute h(4). This involves substituting x = 4 into the expression for h(x). Let's go ahead and perform this calculation.

Step 1: Evaluate h(4)

To find h(4), we simply substitute x = 4 into the function h(x) = x³ + 6x: So, let's plug in 4 wherever we see x in the equation for h(x).

  • h(4) = (4)³ + 6(4)

Now, we just need to do the math. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction. This mnemonic helps us remember the correct sequence of operations, ensuring we arrive at the accurate result. In this case, we first handle the exponent, then multiplication, and finally addition. The correct application of the order of operations is crucial in mathematics, as it ensures consistency and avoids ambiguity in calculations. Without a standardized order, mathematical expressions could be interpreted in multiple ways, leading to incorrect results and confusion. PEMDAS/BODMAS provides this necessary standardization, allowing mathematicians and scientists to communicate effectively and accurately. Let's apply this to our problem and compute the value of h(4).

First, we calculate the exponent: 4 cubed (4³) is 4 * 4 * 4, which equals 64. Exponents represent repeated multiplication and are fundamental in many areas of mathematics, including algebra, calculus, and number theory. Understanding how exponents work is essential for simplifying expressions, solving equations, and analyzing functions. In the context of our problem, the cubic term 4³ contributes significantly to the value of h(4), highlighting the impact of exponents on polynomial functions. Next, we perform the multiplication: 6 times 4 is 24. Multiplication is one of the basic arithmetic operations, and its role here is to scale the input x by a factor of 6. This scaling effect is a common feature in linear terms within polynomial functions, influencing the rate of change of the function. Finally, we add the two results together: 64 + 24 = 88. Addition is the most fundamental arithmetic operation, combining two quantities into a single sum. In the case of h(4), addition brings together the cubic and linear terms, resulting in the final value of the function at x = 4. This sum represents the overall output of the function h for the given input, and it forms the intermediate step in evaluating the composite function. So, we have:

  • h(4) = 64 + 24 = 88

Great! Now we know that h(4) is 88. This is a crucial piece of information, because now we can use it as the input for our next function, g(x). Understanding the value of h(4) is pivotal because it bridges the two functions in our composite function. The output of h(4), which is 88, now becomes the input for the function g(x). This transfer of output to input is the essence of function composition, where the result of one function directly influences the subsequent function. Without accurately calculating h(4), we cannot correctly evaluate (g ∘ h)(4), highlighting the sequential nature of composite functions. So, now that we know h(4) = 88, we are ready to substitute this value into g(x). Let's move on to the next step, where we will evaluate g(h(4)) using this result.

Step 2: Evaluate g(h(4))

Now that we know h(4) = 88, we can plug this into g(x). Remember, g(x) = √(5x). So, we need to find g(88). Evaluating g(h(4)) is the heart of solving the composite function problem. This step requires us to take the output from the function h, which we calculated to be 88, and use it as the input for the function g. This process illustrates the fundamental concept of function composition, where the result of one function acts as the starting point for another. By substituting h(4) into g(x), we are effectively linking the two functions together, creating a new function that describes their combined behavior. The success of this step depends on our accurate evaluation of h(4) and our understanding of how g(x) operates on its input. This step also highlights the importance of following the correct order of operations in composite functions: first, we evaluate the inner function, and then we use that result to evaluate the outer function. This sequential approach ensures that we correctly apply the functions in the intended order, leading to the accurate evaluation of the composite function. To evaluate g(88), we substitute 88 for x in the expression for g(x). Let's perform this substitution and see what we get.

  • g(88) = √(5 * 88)

Okay, let's simplify this. First, we need to multiply 5 and 88. Multiplying 5 by 88 is a straightforward arithmetic operation, but it's essential to perform it accurately to maintain the correctness of our calculation. Multiplication is a fundamental operation in mathematics, and its role here is to scale the input to the square root function. By multiplying 5 and 88, we are determining the value that will be under the square root, which will directly impact the final result. The accuracy of this multiplication is crucial, as any error here will propagate through the rest of the calculation and affect our final answer. To ensure precision, we can use basic multiplication techniques or a calculator to find the product of 5 and 88. This simple multiplication step is a key component in evaluating the composite function, and it sets the stage for the final step of taking the square root. So, let’s make sure we get this part right.

  • 5 * 88 = 440

So, now we have:

  • g(88) = √440

Now, we need to simplify the square root. Simplifying square roots often involves factoring the number under the radical and looking for perfect square factors. This process helps us express the square root in its simplest form, making it easier to work with and interpret. In the case of √440, we need to identify the largest perfect square that divides 440 evenly. This involves breaking down 440 into its prime factors and looking for pairs of identical factors, which can then be taken out of the square root as a single factor. The ability to simplify square roots is a valuable skill in mathematics, as it allows us to express irrational numbers in a more manageable way and perform further calculations with them. This process is not only useful in evaluating composite functions but also in various other areas of mathematics, including algebra, geometry, and calculus. By simplifying √440, we will arrive at a more concise and understandable expression for the value of g(88).

Step 3: Simplify the Square Root

To simplify √440, we need to find the prime factorization of 440. Let's break it down: The prime factorization of a number is expressing it as a product of its prime factors, which are numbers that have only two distinct factors: 1 and themselves. Finding the prime factorization is a crucial step in simplifying square roots because it allows us to identify perfect square factors within the number. These perfect square factors can then be taken out of the square root, simplifying the expression. The process of prime factorization involves repeatedly dividing the number by its smallest prime factor until we are left with only prime numbers. This method ensures that we have identified all the prime factors and their multiplicities, which is essential for accurate simplification. Understanding prime factorization is a fundamental skill in number theory and is widely used in various mathematical applications, including simplifying radicals, finding the greatest common divisor, and solving Diophantine equations. In the context of our problem, prime factorization will help us rewrite √440 in its simplest form, making it easier to compare with the answer choices.

  • 440 = 2 * 220
  • 220 = 2 * 110
  • 110 = 2 * 55
  • 55 = 5 * 11

So, the prime factorization of 440 is 2 * 2 * 2 * 5 * 11. Now we can rewrite the square root: Rewriting the square root using the prime factorization allows us to identify and extract perfect square factors, which simplifies the expression. Perfect squares are numbers that can be expressed as the square of an integer, such as 4 (2²), 9 (3²), and 16 (4²). In our case, we are looking for pairs of identical prime factors in the factorization of 440, as each pair represents a perfect square. By identifying these pairs, we can take their square root and move them outside the radical, leaving the remaining factors inside. This process is based on the property √(a²b) = a√b, which allows us to separate perfect square factors from the rest of the number under the square root. Simplifying square roots in this way is a common practice in mathematics, making expressions easier to understand and work with. It also helps in comparing different expressions and ensuring that answers are in their simplest form.

  • √440 = √(2 * 2 * 2 * 5 * 11)
  • √440 = √(2² * 2 * 5 * 11)
  • √440 = 2√(2 * 5 * 11)
  • √440 = 2√110*

Final Answer

Therefore, (g ∘ h)(4) = 2√110. Looking at our options, this corresponds to answer D. Woohoo! We did it! We successfully evaluated the composite function (g ∘ h)(4) and arrived at the correct answer. This process involved several key steps, each building upon the previous one. First, we defined and understood the concept of composite functions, recognizing that (g ∘ h)(4) means applying h to 4 first, and then applying g to the result. This understanding is crucial for correctly interpreting and solving composite function problems. Next, we evaluated h(4) by substituting 4 into the expression for h(x), which gave us 88. This intermediate step was essential, as the output of h(4) became the input for the function g. Then, we evaluated g(h(4)) by substituting 88 into the expression for g(x), resulting in √440. Finally, we simplified √440 by finding the prime factorization of 440 and extracting perfect square factors, which led us to the simplified form 2√110. This step showcased our ability to work with radicals and simplify them to their simplest form. Each of these steps required careful attention to detail and a solid understanding of mathematical principles. By breaking down the problem into manageable parts and systematically addressing each part, we were able to confidently arrive at the correct answer. This methodical approach is key to success in solving more complex mathematical problems.

Key Takeaways

So, what did we learn today? Evaluating composite functions might seem tricky at first, but by breaking it down into smaller steps, it becomes much more manageable. Remember these key points:

  1. Understand the notation: (g ∘ h)(x) means apply h first, then g.
  2. Work from the inside out: Evaluate the inner function first.
  3. Simplify when possible: Simplify square roots or other expressions to get to the final answer.

Composite functions are a fundamental concept in mathematics, and mastering them opens the door to more advanced topics. They appear in various contexts, from calculus to computer science, and understanding them is essential for building a strong mathematical foundation. The ability to evaluate composite functions allows us to model complex relationships between variables, where the output of one process becomes the input of another. This skill is particularly valuable in fields like engineering, physics, and economics, where systems often involve multiple interconnected processes. Furthermore, working with composite functions enhances our problem-solving skills by requiring us to think systematically and follow a logical sequence of steps. The process of evaluating a composite function involves careful attention to detail and the ability to apply mathematical rules correctly. These skills are transferable to other areas of mathematics and beyond, making the study of composite functions a worthwhile investment in our mathematical education. So, keep practicing, and you'll become a pro at evaluating composite functions in no time! If you guys have any questions, feel free to ask! Keep practicing, and you'll become a master of composite functions in no time!