Function Composition Explained Find (f ⋅ G)(x) When X = 3

Introduction

In the realm of mathematics, particularly in function theory, understanding how functions interact with each other is paramount. One such interaction is the composition of functions, denoted as $(f ext{⋅} g)(x)$, which essentially means applying one function to the result of another. This article delves deep into the concept of function composition, using the specific example of $f(x) = 2x$ and $g(x) = 5x + 2$. We will meticulously explore how to calculate $(f ext{⋅} g)(x)$ and, more importantly, evaluate it when $x = 3$. This exploration will not only solidify your understanding of function composition but also provide a framework for tackling similar problems in the future. Mastering these concepts is crucial for anyone venturing into higher-level mathematics, as function composition forms the bedrock for many advanced mathematical ideas. We will break down each step, ensuring clarity and comprehension for readers of all backgrounds. By the end of this article, you'll be equipped with the knowledge and confidence to handle function composition problems with ease.

Defining Function Composition

Before diving into the specifics of our example, let's first establish a clear understanding of what function composition entails. The composition of two functions, say $f$ and $g$, denoted as $(f ext{⋅} g)(x)$, is defined as $f(g(x))$. This means that we first apply the function $g$ to the input $x$, obtaining an output $g(x)$. Then, we take this output $g(x)$ and use it as the input for the function $f$, resulting in $f(g(x))$. In essence, we are chaining the functions together, where the output of the inner function ($g$ in this case) becomes the input of the outer function ($f$). This process can be visualized as a sequence of operations: $x ightarrow g(x) ightarrow f(g(x))$. The order of functions is crucial in function composition, as $(f ext{⋅} g)(x)$ is generally not the same as $(g ext{⋅} f)(x)$. To illustrate this further, consider a real-world analogy: imagine a machine that first grinds coffee beans (function $g$) and then brews the ground coffee (function $f$). The result (brewed coffee) is different from first brewing something and then grinding it. This analogy highlights the importance of the order of operations in function composition. Understanding this fundamental concept is key to navigating more complex mathematical scenarios involving functions.

Applying Function Composition to the Given Functions

Now, let's apply the concept of function composition to the specific functions given: $f(x) = 2x$ and $g(x) = 5x + 2$. Our goal is to find $(f ext{⋅} g)(x)$, which, as we established earlier, is equivalent to $f(g(x))$. To do this, we need to substitute the function $g(x)$ into the function $f(x)$. This means replacing the 'x' in $f(x)$ with the entire expression for $g(x)$, which is $5x + 2$. So, we have:

$(f ext{⋅} g)(x) = f(g(x)) = f(5x + 2)$.

Now, we apply the function $f$ to the expression $5x + 2$. Since $f(x) = 2x$, we replace 'x' with '$5x + 2$':

$f(5x + 2) = 2(5x + 2)$.

Next, we simplify the expression by distributing the 2:

$2(5x + 2) = 10x + 4$.

Therefore, we have found the composite function: $(f ext{⋅} g)(x) = 10x + 4$. This expression represents the combined effect of applying $g$ and then $f$ to any input $x$. The process of substitution and simplification is a cornerstone of function composition, and mastering it is essential for success in more advanced mathematical topics. We have effectively created a new function that embodies the sequential application of the original two functions. This new function, $10x + 4$, now encapsulates the combined transformation resulting from first applying $g(x)$ and then $f(x)$.

Evaluating (f ⋅ g)(x) when x = 3

Having determined the composite function $(f ext{⋅} g)(x) = 10x + 4$, our next step is to evaluate this function at a specific value, $x = 3$. This means we will substitute $x = 3$ into the expression we derived for $(f ext{⋅} g)(x)$. So, we have:

$(f ext{⋅} g)(3) = 10(3) + 4$.

Now, we perform the arithmetic operations. First, we multiply 10 by 3:

$10(3) = 30$.

Then, we add 4 to the result:

$30 + 4 = 34$.

Therefore, $(f ext{⋅} g)(3) = 34$. This result tells us that when we first apply the function $g$ to the input 3 and then apply the function $f$ to the result, the final output is 34. This process of evaluation highlights the practical application of function composition. We are not just finding a general expression for the composite function but also determining its specific output for a given input. This ability to evaluate composite functions is crucial in various fields, such as physics and engineering, where mathematical models often involve the sequential application of different transformations. The final answer, 34, represents a concrete value that arises from the interplay of the two functions at a specific point.

Importance of Order in Function Composition

To further emphasize the concept of function composition, it's crucial to highlight the significance of the order in which the functions are applied. As we mentioned earlier, $(f ext{⋅} g)(x)$ is generally not the same as $(g ext{⋅} f)(x)$. Let's demonstrate this by finding $(g ext{⋅} f)(x)$ for our given functions, $f(x) = 2x$ and $g(x) = 5x + 2$.

$(g ext{⋅} f)(x)$ is defined as $g(f(x))$. This means we first apply the function $f$ to the input $x$, obtaining $f(x) = 2x$. Then, we take this result, $2x$, and use it as the input for the function $g$. So, we have:

$(g ext{⋅} f)(x) = g(f(x)) = g(2x)$.

Now, we apply the function $g$ to the expression $2x$. Since $g(x) = 5x + 2$, we replace 'x' with '$2x$':

$g(2x) = 5(2x) + 2$.

Next, we simplify the expression:

$5(2x) + 2 = 10x + 2$.

Therefore, $(g ext{⋅} f)(x) = 10x + 2$. Notice that this is different from $(f ext{⋅} g)(x) = 10x + 4$. This difference underscores the importance of order in function composition. Applying functions in a different sequence leads to a different composite function and, consequently, different outputs for the same input. Now, let's evaluate $(g ext{⋅} f)(x)$ when $x = 3$:

$(g ext{⋅} f)(3) = 10(3) + 2 = 30 + 2 = 32$.

Comparing this to our previous result, $(f ext{⋅} g)(3) = 34$, we clearly see that the order of function application matters significantly. This principle is fundamental to understanding function composition and its applications in various mathematical and real-world contexts. The discrepancy in the results highlights that function composition is not commutative; the order of operations directly impacts the outcome.

Conclusion

In this comprehensive exploration, we have delved into the concept of function composition, using the example of $f(x) = 2x$ and $g(x) = 5x + 2$. We learned how to find the composite function $(f ext{⋅} g)(x)$, which involves substituting $g(x)$ into $f(x)$ and simplifying the resulting expression. We then evaluated $(f ext{⋅} g)(x)$ at $x = 3$, demonstrating how to obtain a specific output for a given input. Furthermore, we emphasized the crucial role of order in function composition by comparing $(f ext{⋅} g)(x)$ with $(g ext{⋅} f)(x)$, illustrating that the order in which functions are applied significantly affects the final result. Understanding function composition is not merely an academic exercise; it's a fundamental skill that underpins many advanced mathematical concepts and has practical applications in various fields. By mastering this concept, you equip yourself with a powerful tool for tackling complex problems and gaining a deeper appreciation for the elegance and interconnectedness of mathematics. The ability to compose functions opens doors to a more nuanced understanding of how mathematical transformations interact and influence each other. As you continue your mathematical journey, the principles of function composition will serve as a valuable foundation for exploring more advanced topics and solving real-world problems.