Understanding Domains: Functions F(x) And G(x) Explained

Hey math enthusiasts! Let's dive into the fascinating world of domains! Specifically, we're going to explore the domains of two functions, f(x) and g(x), defined as follows. Knowing how to find a function's domain is super important, so let's get started. Get ready to flex those math muscles and understand this core concept. We'll break down the process step by step, so you can easily master it. By the end, you'll be able to confidently determine the domain of functions like these and many more. This knowledge is fundamental for understanding how functions behave and how to use them effectively. Let's start with the first function, f(x). You'll discover how to identify potential problem areas (like dividing by zero), exclude them from the domain, and express the allowed values in a clear, concise manner. We'll then move on to g(x), applying the same principles to ensure you become proficient in finding the domains of rational functions. So, let's get into it, and you'll be a domain expert in no time!

Function f(x)'s Domain: A Detailed Explanation

Let's get down to the nitty-gritty of finding the domain of the function f(x). Remember, the function f(x) is defined as f(x) = (x + 5) / (x^2 - 10x + 25). The domain of a function refers to all possible input values (x-values) for which the function is defined and produces a valid output. With rational functions (functions that are fractions with polynomials), we have to be extra careful, as we cannot divide by zero. So, our primary goal is to identify any x-values that would cause the denominator to become zero, which would make the function undefined. This is the heart of finding a rational function's domain.

Now, let's focus on the denominator: x^2 - 10x + 25. To find the values of x that make this equal to zero, we need to solve the quadratic equation x^2 - 10x + 25 = 0. This quadratic equation is a perfect square trinomial, which means it can be factored easily. It can be factored into (x - 5)(x - 5) = 0, or (x - 5)^2 = 0. Therefore, the only value of x that makes the denominator zero is x = 5. This is the value that we need to exclude from the domain. So, the domain of f(x) consists of all real numbers except 5. The domain is all the x-values that are allowed to make the equation work.

To express the domain as an interval, we can use the following notation: (-∞, 5) ∪ (5, ∞). This notation means that the domain includes all real numbers from negative infinity up to 5, excluding 5 itself, and then all real numbers from 5 to positive infinity, excluding 5. Essentially, it covers the entire real number line, with a single “hole” at x = 5. You can visualize this on a number line, with an open circle at 5 to indicate that 5 is not included. This interval notation is a clear and concise way to represent the domain, making it easy to understand which values are allowed for the function. Understanding how to exclude these values is critical to fully understanding the function's domain. Understanding domains is a fundamental skill in mathematics, so kudos to you for sticking with me.

Function g(x)'s Domain: Unveiling the Constraints

Okay, let's now switch gears and explore the domain of the function g(x). We have g(x) = (x - 8) / (x^2 - 64). Just like with f(x), we'll be dealing with a rational function. Therefore, the name of the game is to identify any x-values that would cause division by zero. Remember, division by zero is not defined in mathematics, so we must exclude these values from the domain. We need to find the values of x that make the denominator x^2 - 64 equal to zero.

To find these values, we can set the denominator equal to zero and solve for x: x^2 - 64 = 0. This is a difference of squares, which we can factor as (x - 8)(x + 8) = 0. This equation has two solutions: x = 8 and x = -8. These are the values of x that make the denominator zero. Therefore, these are the values that we need to exclude from the domain of g(x). So, the domain of g(x) consists of all real numbers except 8 and -8. It's a key step in understanding a function. Finding where the function is undefined is critical.

When we write the domain in interval notation, we must exclude both of these values. The correct notation for the domain of g(x) is (-∞, -8) ∪ (-8, 8) ∪ (8, ∞). This interval notation breaks down the real number line into three separate intervals: all real numbers less than -8, all real numbers between -8 and 8, and all real numbers greater than 8. Open circles are placed at -8 and 8 on the number line to indicate that these values are not included in the domain. Understanding how to find and represent domains like this is essential for success in higher-level math. This is a very common type of question. You're doing great, keep it up!

Summarizing Domains: A Quick Recap

Alright, let's quickly recap what we've learned about the domains of f(x) and g(x). We have successfully identified and excluded the values of x that would make the denominators of these rational functions equal to zero. This is the critical step in determining the domain.

For f(x) = (x + 5) / (x^2 - 10x + 25), we found that the denominator becomes zero when x = 5. Thus, the domain of f(x) is all real numbers except 5, or (-∞, 5) ∪ (5, ∞). This indicates that the function is defined for all x-values except for x = 5. Remember, this exclusion is because dividing by zero is undefined.

For g(x) = (x - 8) / (x^2 - 64), we found that the denominator becomes zero when x = 8 and x = -8. Therefore, the domain of g(x) is all real numbers except 8 and -8, or (-∞, -8) ∪ (-8, 8) ∪ (8, ∞). This means that the function is defined for all x-values except for x = 8 and x = -8. This exclusion is again due to division by zero.

By following these steps, you can confidently determine the domains of various functions, particularly rational functions. This skill is a foundational element in calculus and other advanced math courses. Keep practicing, and you'll become a pro at finding function domains! Remember to always consider the potential for division by zero and any other restrictions imposed by the function's definition. Well done, guys! You've successfully navigated through finding the domains of these functions and now have a solid understanding of this vital mathematical concept.