Function Composition Explained Solving (f ∘ G)(x) Step-by-Step

$$$$$$

In the realm of mathematical functions, function composition stands as a fundamental operation, allowing us to combine functions and create new ones. This article delves into the intricacies of function composition, guiding you through a step-by-step solution to determine $(f \circ g)(x)$ given the functions $f(x) = -3x^2 - 4\cos x$ and $g(x) = 6x + 4$. We'll break down the process, ensuring clarity and understanding at every stage.

Understanding Function Composition

At its core, function composition involves applying one function to the result of another. The notation $(f \circ g)(x)$ signifies that we first evaluate the function $g(x)$, and then use the output of $g(x)$ as the input for the function $f(x)$. In simpler terms, we're plugging $g(x)$ into $f(x)$. This seemingly straightforward concept forms the bedrock for more advanced mathematical concepts, making it crucial to grasp its mechanics thoroughly.

Defining the Composite Function

The composite function $(f \circ g)(x)$ is formally defined as $f(g(x))$. This notation highlights the sequential nature of the operation: $g(x)$ is evaluated first, and its result becomes the argument for the function $f$. To effectively work with composite functions, it's essential to internalize this order of operations. Think of it as a chain reaction, where the output of one function triggers the action of the next.

Importance of Order

The order in which functions are composed matters significantly. In general, $(f \circ g)(x)$ is not the same as $(g \circ f)(x)$. This non-commutativity underscores the importance of carefully following the notation and understanding which function is applied first. To illustrate, consider simple functions like $f(x) = x + 1$ and $g(x) = 2x$. $(f \circ g)(x) = f(2x) = 2x + 1$, while $(g \circ f)(x) = g(x + 1) = 2(x + 1) = 2x + 2$. The different results highlight the critical role of order in function composition.

Step-by-Step Solution for $(f \circ g)(x)$

Now, let's apply the concept of function composition to the specific functions given: $f(x) = -3x^2 - 4\cos x$ and $g(x) = 6x + 4$. Our goal is to find $(f \circ g)(x)$, which, as we know, is equivalent to $f(g(x))$. This means we need to substitute the expression for $g(x)$ into the function $f(x)$ wherever we see an 'x'.

Step 1: Substitute $g(x)$ into $f(x)$

The first step is to replace the 'x' in $f(x)$ with the entire expression for $g(x)$, which is $6x + 4$. This gives us:

$f(g(x)) = -3(6x + 4)^2 - 4\cos(6x + 4)$

This substitution is the cornerstone of function composition. It's where we bridge the two functions, creating a new function that embodies their combined action. Accuracy in this step is paramount, as any error here will propagate through the rest of the solution. Double-checking the substitution can save significant time and effort in the long run.

Step 2: Expand and Simplify

The next step involves expanding the squared term and simplifying the expression. We need to carefully apply the rules of algebra to ensure we arrive at the correct simplified form. The term $(6x + 4)^2$ requires special attention, as it involves squaring a binomial.

First, let's expand $(6x + 4)^2$:

$(6x + 4)^2 = (6x + 4)(6x + 4) = 36x^2 + 24x + 24x + 16 = 36x^2 + 48x + 16$

Now, substitute this back into our expression for $f(g(x))$:

$f(g(x)) = -3(36x^2 + 48x + 16) - 4\cos(6x + 4)$

Distribute the -3:

$f(g(x)) = -108x^2 - 144x - 48 - 4\cos(6x + 4)$

This expansion is crucial for revealing the polynomial terms and combining like terms. The distributive property plays a vital role here, ensuring that each term within the parentheses is correctly multiplied by the constant outside.

Step 3: Analyze the Result

At this point, we have:

$(f \circ g)(x) = -108x^2 - 144x - 48 - 4\cos(6x + 4)$

This is the simplified expression for $(f \circ g)(x)$. It's important to note that the term $-4\cos(6x + 4)$ cannot be further simplified algebraically. The cosine function's argument is a linear expression, and there are no trigonometric identities that allow for a straightforward simplification in this case. Therefore, our final expression includes both a polynomial part and a trigonometric part.

Examining the Provided Options

Now, let's compare our result with the options provided in the original problem. The options were:

• A. $(f \circ g)(x) = -15x^2 - 16$
• B. $(f \circ g)(x) = -76x^2 - 52$
• C. $(f \circ g)(x) = -75x^2 + 120x + 44$
• D. $(f \circ g)(x) = -108x^2 - 144x - 48 - 4\cos(6x + 4)$

Identifying the Correct Answer

By comparing our derived expression with the options, we can clearly see that option D matches our result:

$(f \circ g)(x) = -108x^2 - 144x - 48 - 4\cos(6x + 4)$

Therefore, option D is the correct answer. This comparison step is crucial in multiple-choice scenarios. It allows us to validate our solution against the available choices, ensuring we select the most accurate representation of our work.

Common Mistakes and How to Avoid Them

Function composition, while conceptually straightforward, can be a source of errors if not approached meticulously. Understanding common pitfalls and how to avoid them is key to mastering this operation.

Incorrect Substitution

One frequent mistake is substituting $f(x)$ into $g(x)$ instead of the other way around. Remember, $(f \circ g)(x)$ means $f(g(x))$, not $g(f(x))$. Always double-check the order of composition to ensure you're substituting the correct function into the other. This involves paying close attention to the notation and internalizing the meaning of the composite function symbol.

Errors in Expansion and Simplification

Another common source of errors lies in the algebraic manipulation of expressions. Expanding squared terms, distributing constants, and combining like terms are all steps where mistakes can easily occur. A careful, methodical approach, with each step written out clearly, can help minimize these errors. Remember the order of operations (PEMDAS/BODMAS) and double-check each expansion and simplification.

Forgetting the Cosine Term

In this particular problem, the presence of the cosine term adds another layer of complexity. Students might sometimes forget to include this term after the substitution, leading to an incomplete and incorrect answer. Always ensure that all parts of the original function are accounted for after the substitution. This often involves careful attention to detail and a systematic approach to the problem.

Misinterpreting Trigonometric Simplification

A final mistake can arise from attempting to simplify the cosine term incorrectly. As we discussed earlier, $-4\cos(6x + 4)$ cannot be simplified further using basic algebraic or trigonometric identities. Recognizing when an expression is in its simplest form is crucial. Avoid forcing simplifications where they don't exist, as this can lead to erroneous results.

Practice Problems to Solidify Understanding

To truly master function composition, practice is essential. Working through a variety of problems helps solidify the concepts and builds confidence in your ability to solve them. Here are a few practice problems to get you started:

1. Let $f(x) = x^2 + 1$ and $g(x) = 2x - 3$. Find $(f \circ g)(x)$ and $(g \circ f)(x)$.
2. Let $f(x) = \sqrt{x}$ and $g(x) = x + 5$. Find $(f \circ g)(x)$ and the domain of the composite function.
3. Let $f(x) = \frac{1}{x}$ and $g(x) = x^2 - 1$. Find $(f \circ g)(x)$ and $(g \circ f)(x)$.

Working through these problems, and others like them, will reinforce your understanding of function composition and help you avoid common errors. Remember to approach each problem systematically, paying close attention to the order of operations and the details of the functions involved.

Conclusion: Mastering Function Composition

Function composition is a fundamental concept in mathematics with far-reaching applications. By understanding the principles of function composition and practicing problem-solving techniques, you can confidently tackle complex mathematical challenges. This article has provided a comprehensive guide to solving $(f \circ g)(x)$ for given functions, highlighting key steps, common mistakes, and strategies for success. As you continue your mathematical journey, remember that consistent practice and a solid understanding of core concepts are the keys to mastery. The ability to manipulate and combine functions opens doors to a deeper understanding of mathematical relationships and problem-solving approaches.

The solution to the problem, $(f \circ g)(x)$ when $f(x) = -3x^2 - 4\cos x$ and $g(x) = 6x + 4$, is indeed:

$(f \circ g)(x) = -108x^2 - 144x - 48 - 4\cos(6x + 4)$

This detailed walkthrough, encompassing the foundational principles, step-by-step solution, error avoidance, and practice problems, equips you with the tools necessary to confidently navigate the world of function composition and beyond. Embrace the challenge, delve into the intricacies, and unlock the power of mathematical functions.