Domain Of F(x) = (x+6) / (x^2 + 4x + 3) Explained

Hey guys! Let's dive into finding the domain of the function F(x) = (x+6) / (x^2 + 4x + 3). This is a classic problem in mathematics that helps us understand where a function is actually defined. We'll break it down step-by-step so it's super clear.

Understanding the Domain of a Function

Before we jump into this specific function, let's quickly recap what the domain actually means. The domain of a function is the set of all possible input values (usually x values) for which the function will produce a valid output. In simpler terms, it's all the x values you can plug into the function without causing any mathematical errors.

For rational functions (like the one we're dealing with), there's one major thing we need to watch out for: division by zero. Division by zero is undefined in mathematics, so any x value that makes the denominator of our function equal to zero must be excluded from the domain.

Identifying Potential Issues

Okay, so with that in mind, let's look at our function again:

F(x) = (x+6) / (x^2 + 4x + 3)

The numerator (x+6) doesn't cause any problems – we can plug any value of x into it. The trouble lies in the denominator: x^2 + 4x + 3. We need to figure out what values of x will make this expression equal to zero. These are the values we'll have to exclude from our domain.

Solving for Undefined Points

To find the values of x that make the denominator zero, we need to solve the following equation:

x^2 + 4x + 3 = 0

This is a quadratic equation, and we can solve it by factoring. Factoring is a method where we rewrite the quadratic expression as a product of two binomials. We're looking for two numbers that multiply to 3 and add up to 4. Those numbers are 3 and 1. So we can factor the equation like this:

(x + 3)(x + 1) = 0

Now, using the zero-product property (which states that if the product of two factors is zero, then at least one of the factors must be zero), we can set each factor equal to zero and solve for x:

x + 3 = 0 or x + 1 = 0

Solving these equations gives us:

x = -3 or x = -1

These are the values of x that will make our denominator zero, meaning they're the values we need to exclude from the domain.

Defining the Domain

Alright, we've found the values that cannot be in our domain. Now, let's state the domain properly. The domain includes all real numbers except for -3 and -1. We can express this in a few different ways:

• Set Notation: {x | x ∈ ℝ, x ≠ -3, x ≠ -1} (This reads: "the set of all x such that x is a real number, and x is not equal to -3, and x is not equal to -1")
• Interval Notation: (-∞, -3) ∪ (-3, -1) ∪ (-1, ∞) (This represents all numbers from negative infinity to -3, then from -3 to -1, and finally from -1 to positive infinity. The "∪" symbol means "union," indicating that we're combining these intervals.)

Interval notation is probably the most common way to express the domain in this context. It clearly shows the intervals where the function is defined.

Visualizing the Domain

It can be helpful to visualize the domain on a number line. Imagine a number line stretching from negative infinity to positive infinity. We'll put open circles at -3 and -1 to indicate that these points are excluded from the domain. The rest of the number line is shaded, showing that all other values are included.

Key Takeaways for Finding the Domain

• Focus on the Denominator: When dealing with rational functions, always check the denominator for values that would make it equal to zero.
• Solve for Undefined Points: Set the denominator equal to zero and solve for x. These are the values you need to exclude.
• Express the Domain Clearly: Use interval notation or set notation to accurately represent the domain.

Why is Understanding Domain Important?

You might be wondering, why does understanding domain even matter? Well, the domain tells us where the function is "well-behaved." It helps us avoid mathematical errors and gives us a more complete picture of the function's behavior. For example, if we were graphing this function, we'd know there would be vertical asymptotes (lines the graph approaches but never touches) at x = -3 and x = -1. Understanding the domain is crucial for many areas of calculus and other advanced math topics.

Let's Practice! - Examples of Finding Domains

To really nail this down, let's look at a few more examples. This will help you see how the same principles apply to different functions.

Example 1: g(x) = 1 / (x - 2)

Okay, first things first, let's identify the denominator: It's (x - 2).

Next, we need to find the values that make the denominator equal to zero. So we set the denominator equal to zero and solve:

x - 2 = 0

Adding 2 to both sides, we get:

x = 2

So, x = 2 is the value we need to exclude from our domain. That's where our function will be undefined.

Now, let's write out the domain in interval notation. We'll have everything from negative infinity up to 2, but not including 2, and then everything from 2 to positive infinity:

Domain: (-∞, 2) ∪ (2, ∞)

See how we use the parentheses? That means we are not including the value 2 in our domain.

Example 2: h(x) = (x + 5) / (x^2 - 9)

Alright, let's tackle this one together! The denominator is (x^2 - 9). This looks a bit more complicated than the last one, but don't worry, we can handle it.

Again, our goal is to find the values of x that make the denominator equal to zero. So let's set it up:

x^2 - 9 = 0

This is a difference of squares, which we can factor into (x + 3)(x - 3). If you remember your factoring rules, this will come to you pretty quickly. If not, it's a good idea to review those!

Now we have:

(x + 3)(x - 3) = 0

Let's use that zero-product property again. We set each factor equal to zero:

x + 3 = 0 or x - 3 = 0

Solving each one, we get:

x = -3 or x = 3

So, we have two values to exclude from our domain this time: -3 and 3.

Let's write the domain in interval notation. We'll have intervals going from negative infinity up to -3, then from -3 to 3, and finally from 3 to positive infinity:

Domain: (-∞, -3) ∪ (-3, 3) ∪ (3, ∞)

Remember those parentheses! They're super important to show that -3 and 3 aren't included.

Example 3: k(x) = √(x + 4)

This one is a bit different! It involves a square root. We can't just look at the denominator this time because there isn't one. However, square roots have their own set of rules about domains.

Remember that we can only take the square root of non-negative numbers (zero or positive numbers). If we try to take the square root of a negative number, we'll end up with an imaginary number, and we're sticking to real numbers for now.

So, what's under the square root (the radicand), which is (x + 4) in this case, must be greater than or equal to zero. Let's write that as an inequality:

x + 4 ≥ 0

Now we just solve for x:

x ≥ -4

This means our domain is all values of x that are greater than or equal to -4. Let's put that in interval notation:

Domain: [-4, ∞)

Notice the square bracket on the -4. That means we are including -4 in the domain because the square root of 0 is perfectly fine.

Conclusion

Finding the domain of a function might seem tricky at first, but with a little practice, you'll get the hang of it. Remember to always watch out for division by zero and square roots of negative numbers. By understanding the domain, you're gaining a deeper understanding of how functions work and where they're defined. Keep practicing, and you'll become a domain-finding pro in no time!