Calculating (f/g)(5) Given F(x) And G(x)

In this article, we will walk through the process of finding the value of $(\frac{f}{g})(5)$ given the functions $f(x) = 7 + 4x$ and $g(x) = \frac{1}{2x}$. This involves understanding function notation, function division, and substitution. By the end of this guide, you'll be able to tackle similar problems with confidence. So, let’s dive in and break down each step to make it crystal clear.

Understanding the Functions

First, let's make sure we understand the functions we're working with. We have two functions:

• $f(x) = 7 + 4x$
• $g(x) = \frac{1}{2x}$

The function $f(x)$ takes an input $x$, multiplies it by 4, and then adds 7. For example, if we input $x = 2$, we get $f(2) = 7 + 4(2) = 7 + 8 = 15$. The function $g(x)$, on the other hand, takes an input $x$, multiplies it by 2, and then takes the reciprocal. So, if we input $x = 2$ into $g(x)$, we get $g(2) = \frac{1}{2(2)} = \frac{1}{4}$. Understanding how these functions operate individually is crucial before we combine them.

Breaking Down Function Notation

Function notation can sometimes seem daunting, but it’s really just a way to express a relationship between an input and an output. When we write $f(x)$, we're saying “$f$ is a function of $x$”. The value inside the parentheses, $x$, is the input, and the function $f$ tells us what to do with that input to get an output. Similarly, $g(x)$ represents a different function that operates on the same input $x$ but performs a different operation. Mastering this notation is key to solving function-related problems.

The Importance of the Domain

Before we proceed, it's important to consider the domain of our functions. The domain is the set of all possible input values for which the function is defined. For $f(x) = 7 + 4x$, there are no restrictions on the input $x$, so the domain is all real numbers. However, for $g(x) = \frac{1}{2x}$, we have a restriction. Since we cannot divide by zero, $x$ cannot be equal to 0. Thus, the domain of $g(x)$ is all real numbers except 0. Keeping the domain in mind is essential, especially when dealing with function division.

Defining (f/g)(x)

Now, let’s understand what $(\frac{f}{g})(x)$ means. This notation represents the division of the function $f(x)$ by the function $g(x)$. In other words:

$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$

To find this, we simply divide the expression for $f(x)$ by the expression for $g(x)$. So, using our given functions, we have:

$(\frac{f}{g})(x) = \frac{7 + 4x}{\frac{1}{2x}}$

Simplifying the Expression

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can simplify the expression as follows:

$(\frac{f}{g})(x) = (7 + 4x) imes (2x)$

Now, we distribute the $2x$ across the terms inside the parentheses:

$(\frac{f}{g})(x) = 2x(7) + 2x(4x)$

$(\frac{f}{g})(x) = 14x + 8x^2$

So, we’ve found that $(\frac{f}{g})(x) = 8x^2 + 14x$. This simplified form makes it much easier to evaluate the function at a specific value.

Domain Considerations for (f/g)(x)

When we divided $f(x)$ by $g(x)$, we created a new function, $(\frac{f}{g})(x)$. It's important to consider the domain of this new function. Initially, $g(x)$ had the restriction that $x \neq 0$. This restriction still applies to $(\frac{f}{g})(x)$, because we cannot divide by zero. Therefore, even though our simplified expression $8x^2 + 14x$ is defined for all real numbers, we must remember that $x \neq 0$ for $(\frac{f}{g})(x)$. Always be mindful of the original domain restrictions when dealing with function operations.

Evaluating (f/g)(5)

Now that we have the simplified expression for $(\frac{f}{g})(x)$, we can find the value of $(\frac{f}{g})(5)$. This means we substitute $x = 5$ into our expression:

$(\frac{f}{g})(5) = 8(5)^2 + 14(5)$

Step-by-Step Calculation

Let’s break down the calculation step by step:

1. First, we calculate $5^2$, which is $5 imes 5 = 25$.
2. Next, we multiply $8$ by $25$, which gives us $8 imes 25 = 200$.
3. Then, we multiply $14$ by $5$, which gives us $14 imes 5 = 70$.
4. Finally, we add the two results together: $200 + 70 = 270$.

So, $(\frac{f}{g})(5) = 270$. This final calculation gives us the answer we were looking for.

Alternative Approach

We could also have found $(\frac{f}{g})(5)$ by first evaluating $f(5)$ and $g(5)$ separately, and then dividing the results. Let's try that:

1. $f(5) = 7 + 4(5) = 7 + 20 = 27$
2. $g(5) = \frac{1}{2(5)} = \frac{1}{10}$

Now, divide $f(5)$ by $g(5)$:

$\frac{f(5)}{g(5)} = \frac{27}{\frac{1}{10}}$

Dividing by $\frac{1}{10}$ is the same as multiplying by 10:

$\frac{27}{\frac{1}{10}} = 27 imes 10 = 270$

As you can see, we get the same result, $(\frac{f}{g})(5) = 270$. This alternative method confirms our previous calculation and demonstrates that there can be multiple ways to solve the same problem.

Conclusion

In this article, we successfully found the value of $(\frac{f}{g})(5)$ given the functions $f(x) = 7 + 4x$ and $g(x) = \frac{1}{2x}$. We started by understanding the functions and their domains, then defined $(\frac{f}{g})(x)$ and simplified it. Finally, we evaluated $(\frac{f}{g})(5)$ using both the simplified expression and by dividing $f(5)$ by $g(5)$. We arrived at the same answer using both methods, which reinforces our understanding of the process. Function operations and evaluations are fundamental concepts in mathematics, and mastering them will greatly enhance your problem-solving skills.

For further exploration and to deepen your understanding of functions, consider visiting resources like Khan Academy's Functions and Equations. This will provide you with a broader perspective and more practice problems to hone your skills.