Domain Of G(x): Interval Calculation Explained

Oct 17, 2025 by ADMIN 47 views

Hey guys! Let's dive into finding the domain of a rational function. It might sound intimidating, but trust me, it's totally doable! We're going to break down a specific example step by step, so you'll be a pro in no time. So, let's get started and make math a little less scary, okay?

Understanding the Domain of a Function

So, what exactly is the domain of a function? Think of it as the set of all possible input values (usually x-values) that you can plug into the function without causing any mathematical mayhem. In simpler terms, it's all the x-values that make the function work! For most functions, like polynomials, the domain is all real numbers – you can plug in anything! But, there are a few situations where we need to be careful, especially with rational functions (fractions with polynomials) and square roots.

When we're dealing with a rational function, which looks like ${ g(x) = \frac{P(x)}{Q(x)} } $, we have to watch out for the denominator, \${ Q(x) \}$ . Why? Because dividing by zero is a big no-no in the math world! It's undefined, and it breaks everything. So, the domain of a rational function is all real numbers except for the values of x that make the denominator equal to zero. These values are called discontinuities or singularities, and they create holes or vertical asymptotes in the graph of the function. Finding these values is crucial for determining the domain. We need to identify the x values that would make the denominator zero and exclude them from the domain. This ensures that our function remains well-defined and doesn't lead to any undefined results. For example, if we have ${ g(x) = \frac{1}{x-2} }$, we know that x cannot be 2, because that would make the denominator zero. So, the domain would be all real numbers except 2. This concept is fundamental in calculus and real analysis, where the behavior of functions around these discontinuities is closely studied. Understanding the domain is the first step in analyzing a function's behavior, graphing it, and solving related problems. So, let's keep this in mind as we move forward and tackle more complex examples!

Our Function: A Quick Look

Okay, let's zoom in on the function we're going to work with today: ${ g(x) = \frac{x^2 + 1}{x^2 - 7x + 12} } $. As you can see, it's a rational function, which means we've got a fraction with polynomials in the numerator and the denominator. The numerator is \${ x^2 + 1 \}$ , which is a simple quadratic expression. It's pretty harmless and doesn't cause any domain issues on its own. The real action is happening in the denominator: ${ x^2 - 7x + 12 } $. This quadratic expression is the key to finding the domain of \${ g(x) \}$ . Remember, we need to figure out what values of x would make this denominator equal to zero, because that's where our function becomes undefined. So, our mission is to solve the equation ${ x^2 - 7x + 12 = 0 }$. Once we find the solutions, we'll know which x-values to exclude from the domain. This is a classic quadratic equation, and there are several ways to solve it, such as factoring, using the quadratic formula, or completing the square. In this case, factoring is probably the easiest method. By understanding the structure of this function, we can focus our attention on the critical part that determines its domain. This kind of analysis is essential in mathematics, where breaking down complex problems into smaller, manageable parts is a common strategy. So, let's move on to the next step and actually solve this quadratic equation to find those troublesome x-values.

Finding the Discontinuities

Alright, let's get our hands dirty and find those pesky x-values that make the denominator zero. We need to solve the equation ${ x^2 - 7x + 12 = 0 } $. As I mentioned earlier, factoring is a neat way to tackle this. We're looking for two numbers that multiply to 12 and add up to -7. Think about it for a sec... what numbers come to mind? If you said -3 and -4, you're spot on! So, we can rewrite the quadratic as \${ (x - 3)(x - 4) = 0 \}$ . Now, this is super helpful because it tells us that the equation is true if either ${ x - 3 = 0 }$ or ${ x - 4 = 0 } $. Solving these simple equations, we get \${ x = 3 \}$ and ${ x = 4 } $. These are the values that make the denominator zero, which means they are the discontinuities of our function. At these points, the function \${ g(x) \}$ is undefined. This also means that the graph of the function will have vertical asymptotes at ${ x = 3 }$ and ${ x = 4 }$. Understanding how to factor quadratic equations is a fundamental skill in algebra and calculus. It allows us to find the roots of the equation, which are crucial for solving many types of problems. Factoring not only simplifies the equation but also gives us a clear understanding of the values that make the expression zero. This technique is widely used in various mathematical contexts, such as solving polynomial equations, simplifying rational expressions, and analyzing the behavior of functions. Now that we've found the values that cause issues, we're ready to define the domain of our function.

Defining the Domain

Okay, we've done the hard work! We know that ${ g(x) }$ is undefined when ${ x = 3 }$ and ${ x = 4 } $. So, the domain of \${ g(x) \}$ is all real numbers except 3 and 4. How do we write that in fancy interval notation? Well, we break it up into intervals that exclude these values. We start from negative infinity and go up to 3, but we don't include 3, so we use a parenthesis: ${ (-\infty, 3) } $. Then, we pick up right after 3 and go up to 4, again excluding 4: \${ (3, 4) \}$ . Finally, we go from 4 to positive infinity: ${ (4, \infty) } $. To put it all together, we use the union symbol \${ \\cup \}$ to join these intervals: ${ (-\infty, 3) \cup (3, 4) \cup (4, \infty) } $. This is the domain of \${ g(x) \}$ in interval notation! So, if we compare this to the given form ${ (-\infty, a) \cup (a, b) \cup (b, \infty) } $, we can see that \${ a = 3 \}$ and ${ b = 4 }$. Interval notation is a concise way to represent sets of numbers, especially when dealing with inequalities and domains of functions. It's a standard notation in mathematics and provides a clear way to express intervals that may include or exclude endpoints. Understanding how to write domains in interval notation is essential for further studies in calculus and analysis, where precise notation is crucial for communicating mathematical ideas. By identifying the discontinuities and expressing the domain in interval notation, we have a complete understanding of the function's valid input values.

Final Answer

So, after all that awesome work, we've found that the domain of ${ g(x) }$ is ${ (-\infty, 3) \cup (3, 4) \cup (4, \infty) } $, and the value of \${ a \}$ is 3. Yay! You've successfully navigated the world of rational functions and domains. You're becoming a math whiz, one problem at a time. Keep up the great work, and remember, math isn't scary when you break it down step by step. Every problem is just a puzzle waiting to be solved, and you've got the skills to do it. So, keep practicing, keep exploring, and most importantly, keep having fun with math!