Find The Domain Of $f(x) = 1/(2x+6)$

Hey guys! Let's dive into a super common math problem that pops up all the time: finding the domain of a function. Specifically, we're gonna tackle this one: f(x)=rac{1}{2 x+6}. Understanding the domain is like knowing the rules of the road for a function – it tells us which input values (the 'x' values) are actually allowed. For rational functions like this one, where we have a fraction with variables in it, the main thing to watch out for is the denominator. You never, ever want to divide by zero, right? That's a big no-no in math, and it'll totally break your function. So, our mission, should we choose to accept it, is to find out which 'x' values would make the denominator equal to zero and then exclude them from our domain. Get ready to flex those math muscles!

So, how do we actually find the values of 'x' that cause trouble? It's pretty straightforward, really. We take that pesky denominator, which is $2x+6$ in our case, and we set it equal to zero. This is like asking, "Hey denominator, when do you decide to bail out and become zero?" So, we write the equation: $2x+6 = 0$. Now, we just solve this simple linear equation for 'x'. First, subtract 6 from both sides to isolate the term with 'x': $2x = -6$. Then, divide both sides by 2 to get 'x' all by its lonesome: x = rac{-6}{2}. And voilà! We get $x = -3$. This little number, -3, is the exact value that makes our denominator zero. Therefore, this value cannot be included in the domain of our function $f(x)$. All other real numbers, however, are perfectly fine inputs. They won't cause any division-by-zero drama. So, the domain includes all real numbers except for -3. It's crucial to remember this distinction: we identify the forbidden value(s) and then state that the domain consists of all other valid numbers.

Now, let's talk about how we write this down in proper mathematical language, specifically using set notation. This is where we formally express the set of all possible input values for our function. The problem gives us a few options, and we need to pick the one that accurately describes our findings. Remember, we figured out that our function f(x)=rac{1}{2 x+6} is defined for all real numbers ($x eq -3$). The standard way to write this in set notation is: {x | x ∈ R, x ≠ -3}. Let's break this down. The curly braces {} indicate that we are defining a set. The 'x' before the vertical bar | means we are talking about the values of 'x'. The vertical bar itself is read as "such that". Then, x ∈ R tells us that 'x' belongs to the set of real numbers (R). Finally, x ≠ -3 is the crucial condition we found – 'x' cannot be equal to -3. So, the entire statement {x | x ∈ R, x ≠ -3} reads as "the set of all x such that x is a real number and x is not equal to -3". This perfectly matches our conclusion. When you look at the options provided (A, B, C, D), you'll see that option A is the one that uses this exact notation and condition. It's super important to get this notation right because it's the standard way mathematicians communicate these sets of values. It’s like speaking the same language, you know?

Let's quickly review why the other options are incorrect, just to really nail this down, guys. Option B suggests the domain is all real numbers except for 3 ({x | x ∈ R, x ≠ 3}). If we were to plug in x = 3 into our original function f(x)=rac{1}{2 x+6}, the denominator would be $2(3) + 6 = 6 + 6 = 12$. Since 12 is not zero, x = 3 is actually a valid input for our function. So, option B is definitely out. Option C states the domain is all real numbers except for -2 ({x | x ∈ R, x ≠ -2}). Let's test x = -2. Plugging it into the denominator gives us $2(-2) + 6 = -4 + 6 = 2$. Again, 2 is not zero, so x = -2 is also a perfectly acceptable input. Therefore, option C is incorrect. Lastly, option D proposes the domain excludes -2 ({x | x ∈ R, x ≠ -2}). Wait, this is the same as C. Let's assume D meant something else, or maybe it's a typo. If it meant to exclude some other number, we'd test it the same way. But as written, it's identical to C and therefore incorrect. The key takeaway here is that we must solve for the value that makes the denominator zero, not just guess or pick numbers that look similar to coefficients or constants in the function. Our careful calculation showed that only x = -3 is the problematic value. So, only option A accurately represents the domain of the function f(x)=rac{1}{2 x+6}. Always trust your calculations, people!

It's super important to be really meticulous when determining the domain, especially with functions involving fractions. The core principle, as we've seen, is to identify any values that would lead to division by zero. For a simple rational function like f(x)=rac{1}{2 x+6}, this means setting the denominator, $2x+6$, to zero and solving for $x$. This gave us $x = -3$. So, the domain is all real numbers except -3. Expressing this in set notation is the standard and most precise way to communicate this. The notation {x | x ∈ R, x ≠ -3} is a concise and universally understood way to represent this set. It clearly defines the set of inputs that are permissible for the function. This means that if you were to graph this function, you'd be able to plug in any real number for $x$ except for -3. At $x = -3$, the function is undefined, and this often manifests as a vertical asymptote on the graph. Understanding the domain is fundamental for analyzing function behavior, solving equations, and much more in mathematics. It's a foundational concept that helps us avoid mathematical errors and fully grasp the capabilities and limitations of a given function. Keep practicing, and you'll become a domain-finding pro in no time, guys!

In conclusion, when faced with a rational function like f(x)=rac{1}{2 x+6}, the process to find its domain is systematic and relies on avoiding division by zero. We isolate the denominator, set it equal to zero, and solve for the variable. In this case, $2x+6 = 0$ yielded $x = -3$. This value must be excluded from the set of all real numbers. The correct representation of this domain in set notation is {x | x ∈ R, x ≠ -3}. This corresponds precisely to option A among the choices provided. Remember this method for any rational function – find the 'x' values that make the denominator zero and exclude them from the set of real numbers. It’s a critical skill, and mastering it will make tackling more complex functions a breeze. Keep up the great work, math enthusiasts!