Dividing Functions Explained How To Find (f/g)(x) When X = 3

In the fascinating world of mathematics, functions are fundamental building blocks that describe relationships between variables. We often encounter scenarios where we need to combine functions through various operations like addition, subtraction, multiplication, and division. This comprehensive guide delves into the concept of dividing functions, providing a clear understanding of the process and illustrating it with a practical example. Specifically, we will explore how to determine $(f extdiv g)(x)$ when given two functions, $f(x)$ and $g(x)$, and evaluate the result at a specific value of $x$. Let's embark on this mathematical journey to unravel the intricacies of function division.

Defining Function Division

The division of functions, denoted as $(f extdiv g)(x)$, is a mathematical operation that combines two functions, $f(x)$ and $g(x)$, by dividing the expression for $f(x)$ by the expression for $g(x)$. Mathematically, it is represented as:

(f extdiv g)(x) = rac{f(x)}{g(x)}

It is crucial to remember that the division of functions is only defined for values of $x$ where the denominator function, $g(x)$, is not equal to zero. This restriction stems from the fundamental principle that division by zero is undefined in mathematics. Therefore, when determining the domain of $(f extdiv g)(x)$, we must exclude any values of $x$ that make $g(x) = 0$.

Before we dive into solving an example, let's emphasize the importance of understanding function notation. The notation $f(x)$ represents a function named "f" that takes "x" as an input. The expression $f(x)$ defines the rule or formula that transforms the input "x" into an output value. Similarly, $g(x)$ represents another function named "g" with its own specific rule. When we divide functions, we are essentially creating a new function whose output is the result of dividing the outputs of the original functions, $f(x)$ and $g(x)$, for the same input value "x."

Step-by-Step Process for Dividing Functions

To effectively divide functions, we can follow a systematic approach:

1. Identify the functions: Begin by clearly identifying the expressions for the functions $f(x)$ and $g(x)$. These expressions will define the rules that govern how the functions transform their inputs into outputs.

2. Write the division: Express the division of the functions as a fraction, placing $f(x)$ in the numerator and $g(x)$ in the denominator:

   (f extdiv g)(x) = rac{f(x)}{g(x)}

3. Substitute the expressions: Replace $f(x)$ and $g(x)$ with their respective algebraic expressions. This step involves substituting the formulas that define the functions into the division expression.

4. Simplify the expression: If possible, simplify the resulting fraction by canceling out common factors or performing algebraic manipulations. Simplification can make the expression easier to understand and work with.

5. Determine the domain: Identify any values of $x$ that would make the denominator, $g(x)$, equal to zero. These values must be excluded from the domain of the resulting function $(f extdiv g)(x)$. The domain represents the set of all valid input values for the function.

6. Evaluate at a specific value (if required): If the problem asks for the value of $(f extdiv g)(x)$ at a specific value of $x$, substitute that value into the simplified expression and calculate the result. This step provides the output of the function division for a particular input.

By following these steps, we can systematically divide functions and gain a comprehensive understanding of the resulting function's behavior.

Illustrative Example: Dividing and Evaluating Functions

Now, let's apply the steps outlined above to a concrete example. Suppose we are given the following functions:

$$f(x) = 2x$$

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

The problem asks us to find $(f extdiv g)(x)$ and then evaluate the result when $x = 3$. Let's proceed step-by-step:

Step 1: Identify the functions

We have already identified the functions:

$$f(x) = 2x$$

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

These functions define the relationships between the input "x" and the corresponding output values.

Step 2: Write the division

Express the division of the functions as a fraction:

(f extdiv g)(x) = rac{f(x)}{g(x)}

This represents the fundamental setup for dividing the functions.

Step 3: Substitute the expressions

Substitute the expressions for $f(x)$ and $g(x)$:

(f extdiv g)(x) = rac{2x}{5x + 2}

This substitution replaces the function notations with their algebraic representations.

Step 4: Simplify the expression

In this case, the expression rac{2x}{5x + 2} cannot be simplified further because there are no common factors to cancel. The numerator and denominator do not share any factors that can be divided out.

Step 5: Determine the domain

To find the domain, we need to identify any values of $x$ that would make the denominator, $5x + 2$, equal to zero. To do this, we set the denominator equal to zero and solve for $x$:

$$5x + 2 = 0$$

$$5x = -2$$

x = -rac{2}{5}

Therefore, the function $(f extdiv g)(x)$ is undefined when x = -rac{2}{5}. The domain of the function is all real numbers except for -rac{2}{5}. In interval notation, this can be expressed as (- ext{∞}, -rac{2}{5}) ext{∪} (-rac{2}{5}, ext{∞}).

Step 6: Evaluate at x = 3

Now, we need to find the value of $(f extdiv g)(x)$ when $x = 3$. Substitute $x = 3$ into the simplified expression:

(f extdiv g)(3) = rac{2(3)}{5(3) + 2}

(f extdiv g)(3) = rac{6}{15 + 2}

(f extdiv g)(3) = rac{6}{17}

Thus, (f extdiv g)(3) = rac{6}{17}. This is the value of the function division when the input is 3.

Final Answer

Therefore, given the functions $f(x) = 2x$ and $g(x) = 5x + 2$, we have found that (f extdiv g)(x) = rac{2x}{5x + 2}, and when $x = 3$, (f extdiv g)(3) = rac{6}{17}. This detailed example demonstrates the process of dividing functions and evaluating the result at a specific point. Remember to always consider the domain of the resulting function to ensure accurate and meaningful results.

In this comprehensive guide, we have explored the concept of dividing functions, outlining a step-by-step process for performing this operation. We emphasized the importance of understanding function notation, simplifying expressions, determining the domain, and evaluating the result at specific values. By working through a detailed example, we solidified our understanding of the process and its practical application. Dividing functions is a valuable tool in mathematics, allowing us to combine and analyze relationships between variables in a more nuanced way. As you continue your mathematical journey, remember the principles and techniques discussed here, and you will be well-equipped to tackle a wide range of function-related problems. The ability to confidently divide functions opens doors to a deeper understanding of mathematical concepts and their applications in various fields.