Finding The Domain: Let's Figure Out $f(x)=\sqrt[6]{5-x}$!

Hey guys! Today, we're diving into a crucial concept in math: finding the domain of a function. Specifically, we're going to crack the code for the function $f(x) = \sqrt[6]{5-x}$. Don't worry, it sounds a bit intimidating at first, but trust me, it's not as scary as it looks! We'll break it down step by step, so you can totally nail this. Understanding the domain is super important because it tells us all the possible $x$ values we can plug into our function without causing any mathematical mayhem. In simpler terms, it's about figuring out what values of $x$ will give us a real number as an output for $f(x)$. So, let's get started and unravel this together!

What Exactly is a Domain, Anyway?

Before we jump into the function, let's make sure we're all on the same page about what the domain actually is. Think of a function like a machine. You put something in (the input, or $x$), and the machine spits out something else (the output, or $f(x)$). The domain is simply the set of all the things you're allowed to put into the machine. It's all the valid inputs. Now, some machines are super versatile and can handle anything you throw at them. But in math, certain functions have rules. They don't like certain inputs because those inputs would break the rules. This is where the concept of the domain comes in. The domain excludes any values of x that will result in undefined results, such as division by zero, or the square root of a negative number. When we talk about finding the domain, we're essentially searching for any potential input values that cause these mathematical problems. This means checking for all sorts of potential issues to restrict the possible $x$ values. The goal is to identify all the acceptable $x$ values.

Now, let's consider a few examples to help you understand better. Consider the function $f(x) = x + 2$. This function can handle any real number. You can plug in 1, 100, -5, or 3.14. It doesn't matter, it works! Therefore, the domain is all real numbers, denoted as $(-\,\infty, \,\infty)$.

On the other hand, the function $g(x) = \frac{1}{x}$ has a different story. If you try to plug in $x = 0$, you'd get division by zero, which is undefined. Therefore, the domain excludes zero and is all real numbers except zero, often written as $(-\infty, 0) \cup (0, \infty)$. See? Different functions, different domains. So, understanding the function itself is key to figuring out the domain. Ready to apply this to our function?

Diving into $f(x)=\sqrt[6]{5-x}$: The Rules of the Game

Alright, let's get down to business and figure out the domain for our function, $f(x) = \sqrt[6]{5-x}$. The key thing to remember is that we're dealing with a sixth root. And here's the golden rule: the even root (like the sixth root, fourth root, or square root) of a negative number is not a real number. It's a complex number. Since we're looking for real numbers for our function's output, we have to make sure that the expression inside the radical (the thing under the root symbol) is not negative. So, what does this mean for us? This means $(5 - x)$ must be greater than or equal to zero.

Here’s a way to think about it: if $(5 - x)$ is negative, then we'd be trying to take the sixth root of a negative number, which isn't allowed in the realm of real numbers. So, we're safe if $(5 - x)$ is zero or positive. It might be helpful to rewrite our function in the following way: $f(x) = (5 - x)^{\frac{1}{6}}$. This way, we can see that we have a power with an even denominator. Thus, the quantity inside the parentheses needs to be greater than or equal to zero. If this expression is equal to zero, we get zero as a solution, and if it is positive, we get a positive number as a solution.

So, our first step in determining the domain is identifying the expression within the radical, which in this case is $(5 - x)$. Next, we need to consider how to restrict this expression to ensure that we will always get a real number when the output for this function is calculated. Because we are taking an even root, we have to restrict the quantity within the radical to be greater than or equal to zero. This is a very important concept to understand. Now let's work through the math to find out what values of x fit the bill.

Solving the Inequality: Finding the Domain

Okay, time to get our hands dirty and actually solve for the domain. We've established that $(5 - x)$ must be greater than or equal to zero. This sets up an inequality for us. Let's write it down:

$5 - x \ge 0$

Our mission is to isolate $x$. Here's how we'll do it step-by-step:

1. Subtract 5 from both sides: This gives us: $-x \ge -5$

2. Multiply or divide by -1: When you multiply or divide both sides of an inequality by a negative number, you have to flip the inequality sign. So, we have: $x \le 5$

And that's it! We've solved for $x$. This inequality tells us that $x$ must be less than or equal to 5. Any value of $x$ that satisfies this inequality is fair game for our function. Any value of $x$ greater than 5 will cause the expression inside the sixth root to become negative, resulting in a complex number.

Expressing the Domain: Putting it all Together

We know that $x \le 5$. We can say with confidence that the domain of our function, $f(x) = \sqrt[6]{5-x}$, is all real numbers less than or equal to 5. Here are a couple of ways to express the domain:

• Interval Notation: $(-\infty, 5]$
• Set Notation: { $x$ | $x \le 5$ }

Both notations mean the same thing. The interval notation, $(-\infty, 5]$, shows that the domain includes all real numbers from negative infinity up to and including 5. The square bracket at 5 indicates that 5 is included in the domain. In set notation, we read { $x$ | $x \le 5$ } as