Unveiling The Domain Of G(x) A Comprehensive Guide
The function g(x) presents an interesting challenge when it comes to its domain. Understanding the domain of a function is crucial in mathematics, as it defines the set of input values for which the function produces a valid output. For the given function, $g(x)=\frac{x^2+4 x-12}{x^2-8 x-9}$, we need to identify any values of $x$ that would lead to an undefined result. Specifically, we must determine the values of $x$ that make the denominator of the fraction equal to zero, since division by zero is undefined in mathematics. To find these excluded values, we will focus on the denominator, $x^2-8x-9$, and set it equal to zero. This will give us a quadratic equation to solve, and the solutions will be the values of $x$ that are not in the domain of $g(x)$. This exploration is fundamental to understanding the behavior and limitations of rational functions.
Identifying Potential Issues in the Domain
When working with rational functions like $g(x)$, it's imperative to consider the denominator. The denominator, in this case, is $x^2-8x-9$. The core principle here is that division by zero is undefined in mathematics. Therefore, any value of $x$ that makes the denominator equal to zero must be excluded from the domain of the function. To pinpoint these values, we set the denominator equal to zero and solve for $x$. This process involves finding the roots of the quadratic equation $x^2-8x-9 = 0$. The roots of this equation represent the x-values that, when substituted into the denominator, result in zero. These values are the key to understanding the domain restrictions of $g(x)$. This step is not just a procedural requirement but a fundamental aspect of ensuring the function is well-defined for all permissible inputs. By excluding these values, we guarantee that the function operates within the bounds of mathematical consistency. This careful consideration of the denominator is a hallmark of working with rational functions and ensures accurate and meaningful results. Understanding this principle is crucial for further analysis of the function's behavior, such as identifying vertical asymptotes and sketching the graph. Ignoring these restrictions can lead to erroneous conclusions about the function's properties.
Solving the Quadratic Equation
To find the values of $x$ that are not in the domain of $g(x)$, we need to solve the quadratic equation $x^2 - 8x - 9 = 0$. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a straightforward approach. We look for two numbers that multiply to -9 and add up to -8. These numbers are -9 and 1. Therefore, we can factor the quadratic equation as $(x - 9)(x + 1) = 0$. Setting each factor equal to zero gives us two possible solutions: $x - 9 = 0$ or $x + 1 = 0$. Solving these linear equations, we find $x = 9$ and $x = -1$. These are the values of $x$ that make the denominator zero, and thus, they are not in the domain of $g(x)$. It's essential to verify these solutions by substituting them back into the original denominator to ensure they indeed result in zero. This step confirms that our solutions are correct and that we have accurately identified the values excluded from the domain. By finding these values, we gain a clear understanding of the function's limitations and can proceed with further analysis, such as determining the function's behavior near these excluded points.
Identifying Values Not in the Domain
After solving the quadratic equation $x^2 - 8x - 9 = 0$, we found two values: $x = 9$ and $x = -1$. These values are critical because they make the denominator of the function $g(x)$ equal to zero. As we've established, division by zero is undefined in mathematics, so these values must be excluded from the domain of $g(x)$. The domain of a function is the set of all possible input values for which the function produces a valid output. In the case of $g(x)$, the domain consists of all real numbers except for 9 and -1. We can express this mathematically as $x \neq 9$ and $x \neq -1$. These restrictions define the limits within which the function can operate meaningfully. It's important to note that these excluded values often correspond to vertical asymptotes on the graph of the function, which are vertical lines that the function approaches but never touches. Understanding the domain and its restrictions is not just a mathematical formality; it's essential for interpreting the function's behavior and its graphical representation. By excluding these values, we ensure that the function remains well-defined and that our analysis is accurate. This careful attention to the domain is a cornerstone of working with rational functions and other types of functions with potential restrictions on their input values.
Expressing the Solution
Having identified the values of $x$ that are not in the domain of $g(x)$, which are 9 and -1, we need to express the solution clearly and concisely. The question asks for these values, separated by commas if there is more than one. Therefore, the solution is simply "9, -1". This format ensures that the answer is easily understood and directly addresses the question's prompt. It's crucial to follow the instructions provided in the question to ensure that the answer is presented in the correct format. In mathematical problem-solving, clarity and precision are paramount. Presenting the solution in the requested format demonstrates a thorough understanding of the problem and the ability to communicate the answer effectively. This seemingly small detail is an integral part of the problem-solving process and reflects a commitment to accuracy and attention to detail. By clearly stating the excluded values, we complete the task of defining the domain restrictions for the function $g(x)$, which is a fundamental step in understanding the function's behavior and properties. This careful expression of the solution reinforces the importance of clear communication in mathematics.
In conclusion, determining the domain of a rational function like $g(x) = \frac{x^2+4 x-12}{x^2-8 x-9}$ involves identifying values that would make the denominator zero. By setting the denominator, $x^2 - 8x - 9$, equal to zero and solving the resulting quadratic equation, we found that $x = 9$ and $x = -1$ are the values that must be excluded from the domain. Therefore, the values of $x$ that are NOT in the domain of $g$ are 9 and -1. This process highlights the importance of understanding the constraints imposed by rational functions and the necessity of careful analysis to ensure accurate and meaningful results. Understanding domain restrictions is a foundational skill in mathematics, particularly in calculus and advanced analysis, where the behavior of functions near points of discontinuity plays a crucial role. By mastering these concepts, we build a solid foundation for more complex mathematical explorations. The ability to identify and address domain restrictions not only ensures the validity of mathematical operations but also provides valuable insights into the characteristics and limitations of functions. This understanding is essential for a comprehensive grasp of mathematical principles and their applications.
