Finding Undefined Values: Domains Of Functions Explained

Hey math enthusiasts! Let's dive into a common challenge in algebra: figuring out the domain of a function and identifying the values that make it undefined. Today, we'll focus on the function g(x) = (x^2 - 7x - 18) / (x^2 - 5x + 4). Our mission? To pinpoint all the x values that are not part of this function's domain. Understanding domains is super important because it tells us what inputs ( x values) are valid for a function and what inputs will cause it to break down. Think of it like this: a function is like a machine. The domain is the set of raw materials that the machine can process. If you feed it something outside of its domain, you'll get an error, or in math terms, an undefined result. In the case of our function, we're dealing with a fraction. And as we all know, dividing by zero is a big no-no in the math world. So, our primary goal is to find the x values that would make the denominator of the fraction equal to zero. These are the troublemakers that we need to exclude from the domain. Let's get started!

To find these values, we need to analyze the denominator of the function g(x). The denominator is x^2 - 5x + 4. Our task is to determine which values of x will result in this expression equaling zero. This involves solving a quadratic equation. This type of equation is a polynomial equation of degree two, meaning the highest power of the variable (in this case, x) is two. It's a fundamental concept in algebra, and the ability to solve it is crucial for a wide range of mathematical and scientific applications. Remember, the solutions to a quadratic equation are the values of x that satisfy the equation, making the quadratic expression equal to zero. To find these values, the first step is often to factor the quadratic expression, if possible. Factoring means breaking down the expression into a product of simpler expressions (usually binomials). If factoring isn't straightforward, we can use other methods such as the quadratic formula. In our case, factoring is possible and it is an efficient way to find the values of x that make the denominator equal to zero. Once we identify these values, we know they are not in the domain of the function, because they lead to division by zero, which is undefined. This step is about isolating the values that are not part of the domain, ensuring that we understand the valid inputs for the function g(x). Let's factor and find those problematic x values.

Factorizing the Denominator

Alright, let's get our hands dirty and factor the denominator: x^2 - 5x + 4. Factoring a quadratic expression involves finding two binomials that, when multiplied together, give us the original quadratic expression. This process often involves some trial and error, but it's a critical skill in algebra. Think of it as detective work, where you're trying to figure out the pieces that make up the whole. In our case, we're looking for two numbers that multiply to give us 4 (the constant term) and add up to -5 (the coefficient of the x term). Let's think through this step by step. We need to find two numbers that fit the bill. After a bit of mental math, we realize that -4 and -1 fit perfectly. (-4) * (-1) = 4, and (-4) + (-1) = -5. This means we can factor the denominator as (x - 4)(x - 1). Now that we have the factored form, it's easier to find the values of x that make the denominator equal to zero. Because we know that the denominator cannot be zero. These are the x values that are excluded from the domain of the function, because they would result in division by zero.

To make sure we get this right, we'll set each factor equal to zero and solve for x. For the factor (x - 4), setting it equal to zero gives us x - 4 = 0. Solving this simple equation, we add 4 to both sides and find that x = 4. This means that when x is 4, the denominator becomes zero, which makes the function undefined. Next, let's look at the factor (x - 1). Setting it equal to zero, we get x - 1 = 0. Adding 1 to both sides, we find that x = 1. This indicates that when x is 1, the denominator also becomes zero, making the function undefined. Therefore, x = 4 and x = 1 are the values that are not in the domain of our function g(x).

Final Answer

So, after all that work, the values of x that are not in the domain of the function g(x) = (x^2 - 7x - 18) / (x^2 - 5x + 4) are x = 1, 4. These are the values that make the denominator zero, leading to an undefined result. Understanding domains and the values excluded from them is a fundamental skill in algebra. It ensures that functions are used correctly and that calculations are valid. You can think of it as a quality control check for math problems.

Remember, in any rational function (a function that is a fraction with polynomials), the values of x that make the denominator equal to zero are always excluded from the domain. Always remember to check for values that make the denominator zero. This simple check ensures that your calculations remain valid and prevents you from running into undefined results. By applying this method, you can confidently determine the domain of any rational function, ensuring you're only using valid inputs. Keep practicing, and you'll become a domain expert in no time! Keep in mind that domains can also be affected by other factors, such as square roots (which require non-negative values under the radical) and logarithms (which require positive values for the argument). This specific example focused on a rational function, but the general principle of identifying undefined values applies to all types of functions. Always be mindful of any operations that might have restrictions on the input values, and you'll be well-equipped to handle various function challenges.

Congratulations, you have mastered the skill of finding the domain and identifying the values that are undefined for a given function. Keep practicing, and you will become proficient at it in no time. This skill is critical for many math problems, so well done!