Domain Of Sqrt(-6x-8): Interval Notation Guide

Hey guys! Today, we're diving into a fun little math problem: finding the domain of the function ${ f(x) = \sqrt{-6x - 8} }$. And to make things even more interesting, we're going to express our answer using interval notation. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step so you can totally nail it. So, let's get started and figure out how to find the domain and represent it in the interval notation. Trust me, by the end of this, you'll be a pro!

Understanding the Domain

Okay, first things first, let's talk about what the domain actually means. In simple terms, the domain of a function is the set of all possible input values (that's the ${ x }$ values) that will produce a real number as an output. Think of it like this: the domain is the list of ${ x }$'s that your function can happily handle without throwing an error or doing something weird. When dealing with square roots, like in our function, we need to be extra careful because we can't take the square root of a negative number and get a real result. That's where the restriction comes in, and it's the key to finding our domain. So, keep this in mind as we move forward – the domain is all about the allowed ${ x }$ values, and for square roots, those values need to ensure we're not square-rooting any negatives!

Why Square Roots Matter

Now, let's zoom in on why square roots are so important when we're figuring out domains. The big thing to remember is that the square root of a negative number isn't a real number. It ventures into the realm of imaginary numbers, which we're not dealing with in this context. So, if we have a function with a square root, like our ${ f(x) = \sqrt{-6x - 8} }$, the expression inside the square root (that's the ${ -6x - 8 }$ part) must be greater than or equal to zero. This is our golden rule for this problem. If the expression inside the square root is positive or zero, we're good to go – the function will give us a real number. But if it's negative, we're in trouble. This restriction is what dictates our domain. We need to find all the ${ x }$ values that make ${ -6x - 8 }$ greater than or equal to zero. This is why understanding square roots is crucial for finding the domain of functions like this one. We're essentially making sure we only plug in ${ x }$ values that keep the math happy and real!

Setting Up the Inequality

Alright, now that we know the deal with square roots and domains, let's get practical. For our function ${ f(x) = \sqrt{-6x - 8} }$, the expression inside the square root is ${ -6x - 8 }$. As we just discussed, this expression must be greater than or equal to zero to get a real number result. So, we can set up an inequality:

${ -6x - 8 \ge 0 }$

This inequality is the key to unlocking our domain. It's saying that whatever ${ x }$ we plug into this expression, the result must be zero or positive. Our next step is to solve this inequality for ${ x }$. This will tell us exactly which ${ x }$ values are allowed in our domain. Think of it like this: we're building a fence around our allowed ${ x }$ values, and the inequality is the blueprint. Solving it is how we figure out where to put the fence posts. So, let's roll up our sleeves and get this inequality solved!

Solving the Inequality

Okay, let's tackle the inequality ${ -6x - 8 \ge 0 }$. Our goal here is to isolate ${ x }$ on one side of the inequality. First, we can add 8 to both sides:

${ -6x - 8 + 8 \ge 0 + 8 }$

This simplifies to:

${ -6x \ge 8 }$

Now, here comes a crucial step. We need to divide both sides by -6 to get ${ x }$ by itself. But remember, when you multiply or divide an inequality by a negative number, you have to flip the inequality sign! So, when we divide by -6, our ${ \ge }$ sign will turn into a ${ \le }$ sign:

${ \frac{-6x}{-6} \le \frac{8}{-6} }$

This simplifies to:

${ x \le -\frac{4}{3} }$

Voila! We've solved for ${ x }$. This inequality tells us that the domain of our function includes all ${ x }$ values that are less than or equal to ${ -\frac{4}{3} }$. This is a major step forward. We now know the range of ${ x }$ values that will keep our function happy and real. The final step is to express this in interval notation, which will give us a neat and tidy way to represent our domain.

Expressing the Domain in Interval Notation

Alright, we've figured out that the domain of our function consists of all ${ x }$ values less than or equal to ${ -\frac{4}{3} }$. Now, let's translate this into interval notation. Interval notation is a cool way to represent a range of numbers using intervals and special symbols. Since our domain includes all numbers less than or equal to ${ -\frac{4}{3} }$, we're talking about a range that starts way down at negative infinity and goes all the way up to ${ -\frac{4}{3} }$.

In interval notation, we use parentheses and brackets to show whether the endpoints are included or not. A parenthesis ${ ( }$ or ${ ) }$ means the endpoint is not included, while a bracket ${ [ }$ or ${ ] }$ means it is. Since our inequality is ${ x \le -\frac{4}{3} }$, which includes ${ -\frac{4}{3} }$, we'll use a bracket on that end. And because we're going all the way to negative infinity, we'll use a parenthesis on that end (since infinity is not a number we can actually reach). So, the domain in interval notation is:

${ (-\infty, -\frac{4}{3}] }$

There you have it! This is the domain of our function ${ f(x) = \sqrt{-6x - 8} }$ expressed in interval notation. It tells us that any ${ x }$ value within this interval will give us a real number when plugged into the function. We've taken it from the initial problem, through the inequality, and landed smoothly in interval notation. Great job!

Visualizing the Interval

Sometimes, seeing a picture can really help solidify our understanding. So, let's visualize this interval on a number line. Imagine a number line stretching out from negative infinity on the left to positive infinity on the right. We're interested in the section of this line that represents our domain, which is ${ (-\infty, -\frac{4}{3}] }$.

First, find ${ -\frac{4}{3} }$ on the number line. That's about -1.33, so it's a bit to the left of -1. Since our interval includes ${ -\frac{4}{3} }$ (the bracket tells us that), we'll put a solid bracket at this point on the number line. This indicates that ${ -\frac{4}{3} }$ is part of our domain.

Now, we need to represent all the numbers less than ${ -\frac{4}{3} }$. That's everything to the left of our bracket on the number line. We can draw a thick line extending from our bracket all the way to the left, indicating that all those numbers are included. And since we're going all the way to negative infinity, we add an arrow at the left end of our line to show that it goes on forever. This visual representation makes it super clear what our domain is: it's all the numbers from negative infinity up to (and including) ${ -\frac{4}{3} }$. Seeing it on a number line can help you connect the interval notation to the actual set of numbers it represents.

Common Mistakes to Avoid

Alright, we've conquered the domain of ${ f(x) = \sqrt{-6x - 8} }$, but let's take a moment to talk about some common pitfalls to watch out for when dealing with these types of problems. Avoiding these mistakes can save you a lot of headaches and help you ace your math assignments!

• Forgetting to Flip the Inequality Sign: This is a big one! Remember, when you multiply or divide an inequality by a negative number, you must flip the inequality sign. If you don't, you'll end up with the wrong range of values for your domain. In our example, we divided by -6, so we had to change ${ \ge }$ to ${ \le }$. Double-check this step whenever you're working with inequalities.

• Incorrectly Interpreting Interval Notation: Interval notation can be a bit tricky at first. Make sure you understand the difference between parentheses and brackets. Parentheses mean the endpoint is not included, while brackets mean it is. Also, remember that infinity always gets a parenthesis because it's not a specific number you can include. Getting the notation wrong can completely change the meaning of your answer.

• Ignoring the Square Root Restriction: The most fundamental mistake is forgetting that the expression inside a square root must be greater than or equal to zero. If you ignore this, you'll include values in your domain that actually make the function undefined. Always start by setting up the inequality that ensures the expression inside the square root is non-negative.

• Algebra Errors: Simple algebra mistakes can throw off your entire solution. Double-check your steps when solving the inequality, especially when adding, subtracting, multiplying, or dividing. A small error early on can lead to a completely wrong answer.

By keeping these common mistakes in mind, you'll be well-equipped to tackle domain problems with confidence and accuracy!

Wrapping Up

So, guys, we've successfully navigated the world of domains and interval notation! We started with the function ${ f(x) = \sqrt{-6x - 8} }$ and, step by step, figured out its domain. We understood why square roots matter, set up the crucial inequality, solved for ${ x }$, and then beautifully expressed our answer in interval notation: ${ (-\infty, -\frac{4}{3}] }$. We even visualized it on a number line and talked about common mistakes to dodge. You've now got a solid understanding of how to find the domain of a function with a square root and how to write it in interval notation.

Remember, the key is to break it down. Understand the restriction imposed by the square root, set up the inequality, solve it carefully (flipping that sign when needed!), and then translate your answer into interval notation. And don't forget to visualize it – the number line can be a real friend! Keep practicing, and you'll become a domain-finding pro in no time. You've got this!