Finding H(g(x)) Given G(x) = 9x - 2 And H(x) = 3x^2 + 2x

In mathematics, particularly in the realm of functions, understanding composite functions is crucial. A composite function is essentially a function that is applied to the result of another function. This article delves into the process of finding the composite function h(g(x)) given two functions, g(x) = 9x - 2 and h(x) = 3x² + 2x. We will break down the steps involved, explain the underlying concepts, and provide a clear, comprehensive guide to solving this type of problem. This topic is fundamental in algebra and calculus, forming the basis for more advanced mathematical concepts. Therefore, a solid grasp of composite functions is essential for anyone pursuing further studies in these fields. We will explore not just the mechanics of substitution but also the logical flow behind it, ensuring a complete understanding of the process. Through detailed explanations and examples, this article aims to make the concept of composite functions accessible and straightforward.

Defining Composite Functions

Before we dive into the specific problem, let's clarify what a composite function is. In simple terms, a composite function is a function that is formed by applying one function to the result of another. If we have two functions, f(x) and g(x), the composite function f(g(x)) (read as "f of g of x") means that we first apply the function g to x, and then we apply the function f to the result. This is a fundamental concept in mathematics, particularly in calculus, where understanding how functions interact with each other is crucial. The order in which the functions are applied matters; f(g(x)) is generally different from g(f(x)). This distinction highlights the importance of carefully analyzing the function composition. We'll see how this works practically with the given functions in the following sections. Understanding the domain and range of these functions is also critical, as it can affect the domain of the composite function. By mastering this concept, you'll be better equipped to tackle more complex mathematical problems involving multiple functions and transformations.

Problem Statement: Finding h(g(x))

Our goal is to express h(g(x)) in terms of x, given the functions g(x) = 9x - 2 and h(x) = 3x² + 2x. This means we need to substitute the entire function g(x) into h(x) wherever we see x. The process involves careful substitution and simplification, ensuring that all terms are correctly accounted for. This is a common type of problem in algebra and calculus, designed to test your understanding of function composition. Successfully solving this requires a systematic approach, paying close attention to the order of operations and algebraic manipulations. We'll break down each step in detail to ensure clarity and accuracy. By the end of this explanation, you should be able to confidently solve similar problems involving composite functions. This skill is not only important for academic success but also has applications in various fields such as engineering, computer science, and economics, where functions are used to model real-world phenomena.

Step-by-Step Solution

To find h(g(x)), we need to substitute g(x) into h(x). Here's how we do it step-by-step:

Step 1: Write out h(x)

First, we write down the expression for h(x), which is given as:

h(x) = 3x² + 2x

This provides the foundation for our substitution. It's crucial to have the original function clearly stated to avoid any errors in the subsequent steps. Writing it out explicitly also helps in visualizing the substitution process, making it easier to understand where the terms will be replaced. This initial step ensures that we have a solid starting point for our calculation, setting the stage for a successful solution. By clearly identifying h(x), we can proceed with confidence to the next step, which involves substituting g(x) into h(x). This systematic approach is essential for solving complex mathematical problems.

Step 2: Substitute g(x) into h(x)

Next, we replace every instance of x in h(x) with the entire expression for g(x), which is 9x - 2. This gives us:

h(g(x)) = 3(9x - 2)² + 2(9x - 2)

This step is the core of finding the composite function. It involves replacing a single variable (x) with a complete function (g(x)). Care must be taken to ensure that the entire expression for g(x) is substituted correctly, including any parentheses or signs. This is where many students can make mistakes, so it's important to double-check this step. The resulting expression, 3(9x - 2)² + 2(9x - 2), now needs to be simplified. This will involve expanding the squared term and distributing the constants, which we'll cover in the next steps. Proper substitution is key to unlocking the rest of the solution.

Step 3: Expand the Expression

Now we need to expand and simplify the expression. Let's start by expanding (9x - 2)²:

(9x - 2)² = (9x - 2)(9x - 2) = 81x² - 18x - 18x + 4 = 81x² - 36x + 4

Then, we distribute the 2 in 2(9x - 2):

2(9x - 2) = 18x - 4

These expansions are crucial for simplifying the composite function. Expanding (9x - 2)² requires careful application of the distributive property or the FOIL method. Each term in the first binomial must be multiplied by each term in the second binomial. Similarly, distributing the 2 involves multiplying it by each term inside the parentheses. These are fundamental algebraic skills that are essential for solving many mathematical problems. Errors in this step can propagate through the rest of the solution, so accuracy is paramount. By expanding these expressions correctly, we set the stage for the final simplification, which will give us h(g(x)) in its simplest form.

Step 4: Substitute the Expanded Terms

Substitute the expanded terms back into the expression for h(g(x)):

h(g(x)) = 3(81x² - 36x + 4) + (18x - 4)

This step brings together the results of our previous expansions. We're now replacing the squared term and the distributed term back into the main equation. It's like piecing together a puzzle, where each part contributes to the final solution. Accuracy is key here to avoid carrying over any mistakes from the previous steps. By substituting the expanded terms, we're one step closer to simplifying the entire expression and expressing h(g(x)) in its simplest form. This process highlights the importance of methodical work in mathematics, where each step builds upon the previous one.

Step 5: Distribute and Simplify

Now, distribute the 3 in the first term:

3(81x² - 36x + 4) = 243x² - 108x + 12

Then, combine like terms:

h(g(x)) = 243x² - 108x + 12 + 18x - 4 = 243x² - 90x + 8

This is the final simplification step. Distributing the 3 involves multiplying it by each term inside the parentheses. Combining like terms means adding or subtracting the coefficients of terms with the same variable and exponent. This step requires careful attention to signs and coefficients to avoid errors. The result, 243x² - 90x + 8, is the expression for h(g(x)) in terms of x. This means we've successfully found the composite function, which is a quadratic function in this case. This skill is valuable in many areas of mathematics and science, where understanding how functions combine is essential for modeling complex systems.

Final Answer

Therefore, the expression for h(g(x)) in terms of x is:

h(g(x)) = 243x² - 90x + 8

This is the culmination of our step-by-step process. We started with the definition of two functions, g(x) and h(x), and through a series of substitutions and simplifications, we arrived at the composite function h(g(x)). The final answer, 243x² - 90x + 8, is a quadratic expression, demonstrating how the composition of functions can result in a function with a different form. This process highlights the power of algebraic manipulation and the importance of understanding function composition. With this final answer, we've successfully addressed the problem statement and provided a clear and comprehensive solution. This knowledge can be applied to similar problems involving different functions and compositions.

Conclusion

In summary, we have successfully found h(g(x)) by substituting g(x) into h(x) and simplifying the resulting expression. The key steps involved writing out h(x), substituting g(x) into h(x), expanding the expression, and simplifying to obtain the final answer: h(g(x)) = 243x² - 90x + 8. This exercise demonstrates the importance of understanding composite functions and the step-by-step process required to solve them. Mastering this skill is crucial for further studies in mathematics, especially in calculus, where function composition is a fundamental concept. The ability to manipulate and simplify algebraic expressions is also essential, as seen in the expansion and combination of terms. By breaking down the problem into smaller, manageable steps, we can tackle complex mathematical challenges with confidence. The process outlined in this article provides a clear roadmap for solving similar problems, ensuring a solid understanding of composite functions and their applications.