How To Find The Domain Of A Rational Function H(x) = (16x + 1) / ((8x + 7)(9x - 2))

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Introduction

In this comprehensive article, we will delve into the process of finding the domain of a rational function. Specifically, we will focus on the function H(x) = (16x + 1) / ((8x + 7)(9x - 2)). Understanding the domain of a function is crucial in mathematics as it defines the set of all possible input values (x-values) for which the function produces a valid output. For rational functions, the primary concern is identifying values that would make the denominator equal to zero, as division by zero is undefined. Therefore, to determine the domain of H(x), we must identify any x-values that cause the denominator, (8x + 7)(9x - 2), to equal zero. This involves setting the denominator to zero and solving for x. The solutions will be the values that are excluded from the domain. By carefully analyzing the function, setting the denominator to zero, and solving for x, we can accurately determine the domain of the rational function H(x). This article aims to provide a step-by-step guide to this process, ensuring a clear understanding of how to find the domain of any rational function.

Understanding Rational Functions and Domains

Before we dive into the specifics of the given function, let's first define what a rational function is and what we mean by its domain. A rational function is any function that can be expressed as the quotient of two polynomials. In simpler terms, it's a function where you have a polynomial divided by another polynomial. Our function, H(x) = (16x + 1) / ((8x + 7)(9x - 2)), clearly fits this description, with (16x + 1) being one polynomial and (8x + 7)(9x - 2) being another.

The domain of a function, on the other hand, is the set of all possible input values (usually x-values) for which the function will produce a valid output. For most polynomial functions, the domain is all real numbers because you can plug in any value for x and get a real number as a result. However, rational functions have a critical exception: we cannot have a zero in the denominator. Division by zero is undefined in mathematics, and any x-value that makes the denominator of a rational function zero must be excluded from the domain. This is because when the denominator is zero, the function's output is undefined, rendering that x-value invalid within the function's context.

Thus, to find the domain of a rational function, we need to identify the values of x that make the denominator equal to zero and exclude them from the set of all real numbers. The remaining set of x-values will constitute the domain of the function. This process is fundamental to understanding the behavior and limitations of rational functions, and it's a critical concept in various fields of mathematics and its applications. In the following sections, we will apply this principle to our specific function, H(x), to determine its domain step-by-step.

Step-by-Step Solution for H(x) = (16x + 1) / ((8x + 7)(9x - 2))

To find the domain of the rational function H(x) = (16x + 1) / ((8x + 7)(9x - 2)), we need to identify the values of x that make the denominator equal to zero. These values will be excluded from the domain.

1. Identify the Denominator

The denominator of the function is (8x + 7)(9x - 2).

2. Set the Denominator Equal to Zero

To find the values of x that make the denominator zero, we set the denominator equal to zero:

(8x + 7)(9x - 2) = 0

3. Solve for x

Since the product of two factors is zero if and only if at least one of the factors is zero, we can set each factor equal to zero and solve for x:

  • 8x + 7 = 0
    • 8x = -7
    • x = -7/8
  • 9x - 2 = 0
    • 9x = 2
    • x = 2/9

Thus, the values x = -7/8 and x = 2/9 make the denominator zero.

4. Exclude the Values from the Domain

Since these values make the denominator zero, they must be excluded from the domain of H(x). The domain of H(x) consists of all real numbers except x = -7/8 and x = 2/9. Therefore, understanding how to solve for these excluded values is essential in determining the valid inputs for the function, ensuring the output remains defined and meaningful within the mathematical context.

Expressing the Domain

Now that we have identified the values that are not in the domain, we need to express the domain in a clear and mathematically accurate way. There are several ways to express the domain, each with its own advantages and conventions.

1. Set Notation

Set notation is a common way to express the domain. We can write the domain of H(x) as follows:

{x | x ∈ ℝ, x ≠ -7/8, x ≠ 2/9}

This notation reads as "the set of all x such that x is a real number, and x is not equal to -7/8, and x is not equal to 2/9." In simpler terms, this means that the domain includes all real numbers except for -7/8 and 2/9. Set notation provides a concise and formal way to define the set of all possible inputs for the function, highlighting the exclusions that ensure the function's validity. This method is particularly useful in theoretical discussions and mathematical proofs, where precision in defining sets is crucial.

2. Interval Notation

Interval notation is another way to express the domain, particularly useful for visualizing the domain on a number line. The domain of H(x) can be expressed in interval notation as follows:

(-∞, -7/8) ∪ (-7/8, 2/9) ∪ (2/9, ∞)

This notation indicates that the domain consists of three intervals: all numbers less than -7/8, all numbers between -7/8 and 2/9, and all numbers greater than 2/9. The symbol "∪" represents the union of these intervals, meaning that the domain includes all numbers in any of these intervals. Interval notation provides a graphical representation of the domain, making it easier to visualize the set of valid inputs on a number line. This method is particularly useful in calculus and real analysis, where understanding intervals and their properties is essential. Furthermore, the interval notation concisely captures the domain's continuous nature, illustrating that all values between the excluded points are valid inputs for the function.

3. Number Line Representation

We can also represent the domain graphically on a number line. Draw a number line and mark the points -7/8 and 2/9. Place open circles at these points to indicate that they are not included in the domain. Shade the regions to the left of -7/8, between -7/8 and 2/9, and to the right of 2/9. This shaded region represents the domain of H(x). The number line representation provides a visual aid that can enhance understanding, especially for those who benefit from visual learning. By shading the regions corresponding to the domain, it becomes immediately clear which values are included and which are excluded. This method is particularly helpful in introductory courses and when dealing with inequalities, as it offers a tangible way to grasp the concept of valid inputs for a function.

Common Mistakes to Avoid

When finding the domain of a rational function, there are several common mistakes that students often make. Being aware of these pitfalls can help ensure accuracy and a deeper understanding of the concept.

1. Forgetting to Exclude Values That Make the Denominator Zero

The most common mistake is forgetting to identify and exclude the values of x that make the denominator equal to zero. As we've emphasized, division by zero is undefined, so these values cannot be part of the domain. Always remember to set the denominator equal to zero and solve for x. Neglecting to do this can lead to an incorrect domain, and consequently, a misunderstanding of the function's behavior. It's a crucial step that forms the foundation of determining the domain for any rational function.

2. Incorrectly Solving for x

Another common mistake is making errors while solving the equation when the denominator is set to zero. This can involve algebraic errors, such as incorrect factoring or misapplication of the quadratic formula. It is essential to carefully check each step in the solution process to avoid these errors. Double-checking the arithmetic and algebraic manipulations can prevent incorrect exclusions from the domain. Precise solving ensures that only the necessary values are excluded, maintaining the integrity of the domain and the function's validity.

3. Including Values That Make Both the Numerator and Denominator Zero

In some cases, a value of x might make both the numerator and the denominator zero. While such a value still makes the denominator zero and should be excluded from the general domain, it's worth noting that these points can sometimes lead to removable discontinuities (holes) in the graph of the function. These are distinct from asymptotes, where the function tends to infinity. Understanding the difference between holes and asymptotes is crucial for a comprehensive analysis of the function's behavior. While these values are not in the domain, the behavior around these points requires special attention in calculus and advanced mathematics.

4. Incorrectly Expressing the Domain

Finally, mistakes can be made when expressing the domain in set notation or interval notation. Ensure that you correctly use the appropriate symbols and conventions. For example, use open intervals (parentheses) for values that are not included in the domain and closed intervals (brackets) for values that are included. Pay attention to the union symbol (∪) when combining intervals. Inaccurate notation can misrepresent the domain, leading to misunderstandings and errors in further analysis. Precision in expressing the domain is as important as finding the correct values to exclude, as it communicates the accurate scope of the function's valid inputs.

Conclusion

In conclusion, finding the domain of the rational function H(x) = (16x + 1) / ((8x + 7)(9x - 2)) involves identifying the values of x that make the denominator equal to zero and excluding them from the set of all real numbers. By setting the denominator (8x + 7)(9x - 2) equal to zero, we found that x = -7/8 and x = 2/9 are the values that must be excluded. Therefore, the domain of H(x) is all real numbers except -7/8 and 2/9.

We can express this domain in several ways:

  • Set notation: {x | x ∈ ℝ, x ≠ -7/8, x ≠ 2/9}
  • Interval notation: (-∞, -7/8) ∪ (-7/8, 2/9) ∪ (2/9, ∞)
  • Number line representation: A number line with open circles at -7/8 and 2/9, and the regions to the left of -7/8, between -7/8 and 2/9, and to the right of 2/9 shaded.

It is essential to understand the concept of the domain and how to find it for rational functions to avoid division by zero and ensure accurate mathematical analysis. By following a systematic approach and avoiding common mistakes, you can confidently determine the domain of any rational function. Understanding the domain is a fundamental aspect of function analysis, enabling a deeper comprehension of a function's behavior, limitations, and applicability in various mathematical contexts.