Finding The Domain Of F(x) = (x+1)/(x^2 - 6x + 8)

Before we dive into this specific problem, let's make sure we're all on the same page about what the domain of a function actually means. Guys, in simple terms, the domain is like the function's playground. It's the set of all possible x-values that you can plug into the function without causing any mathematical mayhem. Think of it as the function's input zone. We need to avoid situations that lead to undefined results, like dividing by zero or taking the square root of a negative number (at least in the realm of real numbers!). So, when we find the domain, we're basically figuring out which x-values are safe to use.

In mathematical speak, the domain of a function f(x) is the set of all real numbers x for which f(x) is defined. This means we need to identify any values of x that would make the function undefined. The most common culprits for causing undefined behavior are:

1. Division by zero: A fraction is undefined when the denominator is zero. So, we need to find any x-values that make the denominator equal to zero and exclude them from the domain.
2. Square roots of negative numbers: The square root of a negative number is not a real number. So, if a function involves a square root, we need to make sure that the expression inside the square root is non-negative.
3. Logarithms of non-positive numbers: The logarithm of a non-positive number (zero or negative) is undefined. So, if a function involves a logarithm, we need to ensure that the argument of the logarithm is positive.

For our specific function, f(x) = (x+1)/(x^2 - 6x + 8), we need to focus on the first case: division by zero. The numerator, (x+1), is perfectly well-behaved for all real numbers. But the denominator, x^2 - 6x + 8, could potentially be zero for some values of x. That's where we need to investigate further. Let's find those pesky values that make the denominator zero and kick them out of our function's playground!

Okay, let's get down to business and find the domain of our function, f(x) = (x+1)/(x^2 - 6x + 8). As we discussed, the key here is to identify any x-values that would make the denominator equal to zero. Why? Because division by zero is a big no-no in the world of mathematics. It leads to an undefined result, and we want our function to be well-defined for all values in its domain.

The denominator of our function is x^2 - 6x + 8. So, what we need to do is figure out when this expression equals zero. In other words, we need to solve the quadratic equation:

x^2 - 6x + 8 = 0

There are a couple of ways we can tackle this. One common method is factoring. We're looking for two numbers that multiply to 8 and add up to -6. Can you think of them? Yep, they're -2 and -4! So, we can factor the quadratic expression as follows:

(x - 2)(x - 4) = 0

Now, we have a product of two factors that equals zero. This means that at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:

x - 2 = 0 => x = 2 x - 4 = 0 => x = 4

Aha! We've found the culprits! The denominator is equal to zero when x = 2 and when x = 4. This means that these two values are the troublemakers that we need to exclude from the domain. If we plug in x = 2 or x = 4 into our function, we'll end up dividing by zero, which is a mathematical no-go zone.

So, what does this mean for the domain of our function? It means that the domain consists of all real numbers except for 2 and 4. We can't include these values because they make the function undefined. In mathematical notation, we can express this domain in a few different ways:

• Set-builder notation: {x | x ∈ ℝ, x ≠ 2, x ≠ 4}
• Interval notation: (-∞, 2) ∪ (2, 4) ∪ (4, ∞)

Both of these notations tell us the same thing: the domain includes all real numbers less than 2, all real numbers between 2 and 4, and all real numbers greater than 4. We use the union symbol (∪) to combine these intervals. The parentheses indicate that we're not including the endpoints (2 and 4) in the intervals.

Now that we've determined the domain of our function, f(x) = (x+1)/(x^2 - 6x + 8), let's take a look at the answer choices provided and see which one matches our findings. Remember, we found that the domain consists of all real numbers except for 2 and 4.

The answer choices were:

A. all real numbers B. all real numbers except -1 C. all real numbers except -4 and -2 D. all real numbers except 2 and 4

Let's analyze each choice:

• A. all real numbers: This is incorrect because we know that 2 and 4 are not in the domain.
• B. all real numbers except -1: This is also incorrect. While -1 might be a value that makes the numerator zero, it doesn't affect the denominator and therefore doesn't cause the function to be undefined. Remember, we're focused on values that make the denominator zero.
• C. all real numbers except -4 and -2: This is incorrect as well. -4 and -2 are not the values that make our denominator zero. We found those to be 2 and 4.
• D. all real numbers except 2 and 4: This is the correct answer! This choice perfectly matches our findings. We determined that the function is undefined when x = 2 and x = 4, so these values must be excluded from the domain.

Therefore, the correct answer is D. all real numbers except 2 and 4. We've successfully navigated the world of domains and found the playground for our function!

So, we've figured out the domain of f(x) = (x+1)/(x^2 - 6x + 8), but you might be wondering, “Why is this even important?” Guys, understanding the domain of a function is absolutely crucial in mathematics and its applications. It's not just a technicality; it's about making sure we're working with valid and meaningful results.

Here's why the domain matters:

• Function Definition: The domain is an integral part of the definition of a function. A function isn't fully defined until we know its domain. It tells us the set of inputs for which the function produces a valid output. Without knowing the domain, we don't have a complete picture of the function's behavior.
• Avoiding Undefined Results: As we've seen, certain operations, like division by zero, lead to undefined results. Identifying the domain allows us to avoid these situations and ensure that our calculations are meaningful. We don't want to be plugging in values that cause our function to blow up!
• Graphing Functions: The domain plays a critical role in graphing functions. When we graph a function, we only plot points for x-values that are within the domain. If we ignore the domain, we might end up plotting points that don't actually belong to the function, leading to an incorrect graph. The graph will have breaks or asymptotes at the values excluded from the domain.
• Real-World Applications: In many real-world applications, functions model physical phenomena or relationships. The domain often represents physical constraints or limitations. For example, if a function models the height of an object over time, the domain might be restricted to positive time values. Understanding the domain ensures that our mathematical model aligns with the real-world situation.
• Calculus and Beyond: The concept of domain is fundamental in calculus and more advanced mathematics. Many calculus operations, like differentiation and integration, rely on the function being defined over a specific interval. A solid understanding of domains is essential for success in these areas.

In essence, the domain tells us where a function is