Finding Composite Functions A Step-by-Step Guide For F(x) = √(2x) + 5 And G(x) = 3x - 4

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In the captivating realm of mathematics, function composition stands as a fundamental operation, allowing us to weave together functions to create new and intricate expressions. This comprehensive guide delves into the process of finding the composite function (f ∘ g)(x), where f(x) = √(2x) + 5 and g(x) = 3x - 4. We will embark on a step-by-step journey, meticulously unraveling the concept of function composition and equipping you with the tools to confidently tackle such problems.

Understanding Function Composition: The Essence of Combining Functions

At its core, function composition involves applying one function to the result of another. Imagine it as a chain reaction, where the output of one function becomes the input for the next. The notation (f ∘ g)(x) signifies that we first apply the function g(x) to the input x, and then take the result and apply the function f to it. In essence, we are plugging the entire function g(x) into the function f(x).

This operation allows us to construct complex functions from simpler ones, revealing deeper relationships and patterns within mathematical expressions. Mastering function composition is crucial for advanced mathematical concepts, including calculus and differential equations. Let us dive into the step-by-step process of finding (f ∘ g)(x) for our given functions.

Step 1: Defining the Functions and the Composition

Before we embark on the computation, let's clearly define the functions at play. We have:

  • f(x) = √(2x) + 5
  • g(x) = 3x - 4

Our mission is to find (f ∘ g)(x), which, as we established earlier, means we need to find f(g(x)). This notation emphasizes that we will substitute the entire expression for g(x) into the function f(x) wherever we see the variable 'x'. Let's proceed to the substitution step to bring this concept to life.

Step 2: The Substitution: Plugging g(x) into f(x)

This is the heart of the function composition process. We replace every instance of 'x' in f(x) with the entire expression for g(x). This yields:

f(g(x)) = √(2(3x - 4)) + 5

Notice how the (3x - 4) from g(x) has taken the place of 'x' within the square root in f(x). This substitution is the cornerstone of function composition, effectively linking the two functions together. Now, our next step involves simplifying the resulting expression to arrive at the final form of (f ∘ g)(x).

Step 3: Simplifying the Expression: Unveiling the Composite Function

Now that we have performed the substitution, the next step is to simplify the expression √(2(3x - 4)) + 5. Let's break down the simplification process:

  1. Distribute the 2 inside the square root: √(6x - 8) + 5

  2. Observe that the expression inside the square root, 6x - 8, cannot be simplified further without factoring. Since there is no perfect square factor, we leave it as is.

  3. The + 5 outside the square root remains untouched as it is a constant term.

Therefore, the simplified expression for (f ∘ g)(x) is:

(f ∘ g)(x) = √(6x - 8) + 5

This is our final composite function, the result of applying g(x) and then f(x) to the input x. It represents a new function derived from the interplay between f and g. Now, let's discuss the implications of this result and its connection to the original functions.

Examining the Result: Connecting the Composite Function to f(x) and g(x)

Our result, (f ∘ g)(x) = √(6x - 8) + 5, reveals the combined effect of the two functions. Notice how the structure of g(x) (3x - 4) is embedded within the square root of the composite function, and the + 5 from f(x) remains as a constant term. This showcases the chain-like nature of function composition.

The domain of the composite function is determined by the domains of both f(x) and g(x), with an added consideration: the output of g(x) must be within the domain of f(x). In our case, the domain of g(x) is all real numbers, but the domain of f(x) is x ≥ 0 (due to the square root). Therefore, we need 6x - 8 ≥ 0, which implies x ≥ 4/3. This restriction ensures that we are taking the square root of a non-negative number.

Function composition is a powerful tool in mathematics, and understanding its mechanics is crucial for more advanced topics. Let's consider some applications of function composition to solidify our understanding.

Applications of Function Composition: Real-World Connections

Function composition finds its way into various real-world scenarios, showcasing its practical relevance. Here are a few examples:

  1. Price Discounts: Imagine a store offering a discount of 20% on all items, followed by an additional $10 off for loyalty members. We can represent these discounts as functions. Let g(x) = 0.8x represent the 20% discount (multiplying the original price x by 0.8), and let f(x) = x - 10 represent the $10 discount. Then (f ∘ g)(x) would represent the final price after both discounts are applied.

  2. Currency Conversion: Converting currency often involves multiple steps. For instance, converting US dollars to Euros might involve first converting to another currency and then to Euros. Each conversion can be represented as a function, and the overall conversion is a composite function.

  3. Manufacturing Processes: In manufacturing, the production of a product often involves a series of steps, each represented by a function. For example, one function might represent the cost of raw materials, another the cost of labor, and a third the cost of packaging. The composite function would then represent the total cost of producing the product.

These examples highlight the versatility of function composition in modeling real-world phenomena. Now, let's solidify our understanding by exploring some common pitfalls to avoid when working with composite functions.

Common Pitfalls to Avoid: Mastering the Nuances of Composition

While function composition is a powerful tool, it's essential to be aware of common errors that can occur. Here are a few pitfalls to avoid:

  1. Incorrect Order of Operations: Remember that (f ∘ g)(x) means applying g(x) first, then f(x). The order matters! Switching the order can lead to drastically different results. (g ∘ f)(x) is generally not the same as (f ∘ g)(x).

  2. Forgetting to Substitute Correctly: Ensure that you replace every instance of 'x' in the outer function with the entire expression for the inner function. A partial substitution can lead to an incorrect composite function.

  3. Ignoring Domain Restrictions: Always consider the domains of both the original functions and the composite function. The domain of the composite function may be restricted by the domains of its constituent functions.

  4. Oversimplification: While simplification is crucial, avoid oversimplifying expressions prematurely. Simplify step by step to minimize the risk of errors.

By being mindful of these pitfalls, you can confidently navigate the world of function composition and avoid common mistakes. Let's now summarize the key takeaways from our exploration.

Key Takeaways: Mastering Function Composition

In this comprehensive guide, we have explored the fascinating concept of function composition. We learned that:

  • Function composition involves applying one function to the result of another.
  • The notation (f ∘ g)(x) signifies applying g(x) first, then f(x).
  • Finding (f ∘ g)(x) involves substituting the entire expression for g(x) into f(x).
  • Simplification is crucial to obtain the final composite function.
  • The domain of the composite function is influenced by the domains of both original functions.
  • Function composition has diverse real-world applications.
  • Avoiding common pitfalls is essential for accurate results.

By mastering these key takeaways, you will be well-equipped to tackle function composition problems with confidence and precision. Now, let's return to the original question and select the correct answer.

Returning to the Original Question: Selecting the Correct Answer

We were tasked with finding (f ∘ g)(x) for f(x) = √(2x) + 5 and g(x) = 3x - 4. Through our step-by-step process, we arrived at the solution:

(f ∘ g)(x) = √(6x - 8) + 5

Therefore, the correct answer is C. (f ∘ g)(x) = √(6x - 8) + 5. This reinforces the power of our systematic approach to function composition. With a solid understanding of the underlying principles and careful execution, you can confidently navigate these mathematical challenges.

In conclusion, function composition is a fundamental concept in mathematics with far-reaching applications. By mastering the process of combining functions, you unlock a powerful tool for problem-solving and mathematical exploration. Remember to focus on the order of operations, substitute carefully, and consider domain restrictions. With practice and a clear understanding of the underlying principles, you can confidently navigate the world of function composition and unravel the intricate relationships between mathematical expressions.