Domain Of F(x) = (x+1)/(x^2-6x+8) Explained!

Hey guys! Today, we're diving into the fascinating world of functions, specifically focusing on how to find the domain of a function. The domain of a function, in simple terms, is the set of all possible input values (x-values) for which the function produces a valid output. We'll be tackling the function f(x) = (x+1)/(x^2 - 6x + 8). This might seem a little intimidating at first, but trust me, we'll break it down step-by-step and make it super clear. So, grab your thinking caps, and let's get started!

Understanding Domains: The Basics

Before we jump into the specifics of our function, let's solidify our understanding of what a domain actually is. Think of a function like a machine: you put something in (the input, or x-value), and the machine spits something out (the output, or f(x) value). The domain is essentially the list of all the things you can put into the machine without breaking it. In the realm of real numbers, there are a few key things that can cause our function machine to break down:

1. Division by zero: This is a big no-no in mathematics. We can't divide any number by zero, as it leads to an undefined result. So, any x-value that makes the denominator of a fraction equal to zero must be excluded from the domain.
2. Square roots of negative numbers: In the world of real numbers, we can't take the square root of a negative number. This would result in an imaginary number, which we're not dealing with in this context. Therefore, any x-value that makes the expression inside a square root negative must also be excluded from the domain.
3. Logarithms of non-positive numbers: Logarithms are only defined for positive numbers. We can't take the logarithm of zero or a negative number. So, any x-value that results in a non-positive argument for a logarithm must be excluded from the domain.

For our function, f(x) = (x+1)/(x^2 - 6x + 8), we have a fraction. This means we need to be particularly mindful of division by zero. The numerator, (x+1), doesn't pose any problems, as it can take any real value. However, the denominator, (x^2 - 6x + 8), is where the potential trouble lies. Our mission is to find out which x-values make this denominator equal to zero, and then exclude those values from our domain. This ensures our function remains well-defined and doesn't lead to mathematical chaos.

Finding the Culprits: Zeros of the Denominator

Okay, now let's get our hands dirty and find those pesky x-values that make the denominator zero. We have the quadratic expression x^2 - 6x + 8. To find its zeros (the values of x that make the expression equal to zero), we need to solve the equation:

x^2 - 6x + 8 = 0

There are a couple of ways we can tackle this. One common method is factoring. Factoring involves breaking down the quadratic expression into two binomials (expressions with two terms) that multiply together to give us the original quadratic. Let's see if we can factor this one. We need to find two numbers that:

• Multiply to give us 8 (the constant term)
• Add up to give us -6 (the coefficient of the x term)

After a little thought, we can see that the numbers -2 and -4 fit the bill perfectly. (-2) * (-4) = 8, and (-2) + (-4) = -6. So, we can factor our quadratic expression as follows:

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

Now, we have a product of two factors that equals zero. The only way this can be true is if at least one of the factors is equal to zero. This gives us two possibilities:

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

So, we've found our culprits! The values x = 2 and x = 4 make the denominator of our function equal to zero. This means our function is undefined at these points. These are the values we need to exclude from our domain.

Defining the Domain: What's Left?

Now that we've identified the problem areas, we can finally define the domain of our function. We know that our function f(x) = (x+1)/(x^2 - 6x + 8) is defined for all real numbers except for the values that make the denominator zero, which are x = 2 and x = 4. So, the domain consists of all real numbers excluding 2 and 4.

There are a few ways we can express this mathematically:

• Set notation: {x | x ∈ ℝ, x ≠ 2, x ≠ 4}. This reads as "the set of all x such that x is a real number, and x is not equal to 2, and x is not equal to 4."
• Interval notation: (-∞, 2) ∪ (2, 4) ∪ (4, ∞). This notation uses intervals to represent ranges of numbers. The symbol "∪" represents the union of sets, meaning we combine the intervals. This notation indicates all numbers from negative infinity up to 2 (but not including 2), combined with all numbers between 2 and 4 (but not including 2 and 4), combined with all numbers from 4 to positive infinity (but not including 4).

Both of these notations accurately describe the domain of our function. The interval notation is often preferred for its conciseness and clarity, especially when dealing with more complex domains.

Putting It All Together: The Answer

So, after our journey through the world of functions and domains, we've arrived at the answer! The domain of the function f(x) = (x+1)/(x^2 - 6x + 8) is all real numbers except 2 and 4. This corresponds to answer choice D: all real numbers except 2 and 4.

We successfully identified the values that make the denominator zero, excluded them from the set of all real numbers, and expressed the domain in both set notation and interval notation. Great job, guys! You've now conquered a fundamental concept in mathematics.

Why Domains Matter: The Bigger Picture

Understanding domains isn't just about solving problems; it's about grasping the fundamental nature of functions and their behavior. The domain tells us where a function is valid and where it's not. This knowledge is crucial for various applications, including:

• Graphing functions: Knowing the domain helps us understand where the graph of a function exists and where it might have breaks or asymptotes (lines that the graph approaches but never touches).
• Calculus: The domain is essential for many calculus operations, such as finding derivatives and integrals. We need to ensure that the operations are performed within the function's valid domain.
• Real-world modeling: Many real-world phenomena can be modeled using functions. Understanding the domain of these functions helps us interpret the model's results realistically. For example, if a function models the population of a species, the domain might be restricted to non-negative numbers, as you can't have a negative population.

In essence, the domain provides a framework for working with functions. It helps us avoid mathematical errors and interpret results in a meaningful way. So, mastering this concept is a significant step towards a deeper understanding of mathematics and its applications.

Practice Makes Perfect: Further Exploration

Now that you've grasped the basics of finding the domain, it's time to put your knowledge to the test! Try finding the domains of the following functions:

1. g(x) = 1/(x - 3)
2. h(x) = √(x + 2)
3. k(x) = ln(x - 1) (where ln represents the natural logarithm)

Remember to consider the potential pitfalls: division by zero, square roots of negative numbers, and logarithms of non-positive numbers. By practicing with different functions, you'll solidify your understanding and become a domain-finding pro!

And that's a wrap, folks! We've explored the concept of domains, tackled a specific function, and discussed why domains are so important. Keep practicing, keep exploring, and keep those mathematical gears turning!