Domain And Range Of Y=3√x A Step-by-Step Explanation

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on how to identify the domain and range. Our spotlight is on the function $y = 3\sqrt{x}$. If you've ever felt a bit puzzled about domains and ranges, don't worry! We're going to break it down in a way that's super easy to understand. So, let's get started and unravel the mysteries of this function!

What are Domain and Range, Anyway?

Before we jump into the specifics of our function, let's quickly recap what domain and range actually mean. Think of a function like a machine. You feed it an input (the x-value), and it spits out an output (the y-value).

• The domain is the set of all possible inputs (x-values) that you can feed into the machine without causing it to break down – in mathematical terms, without leading to undefined results like division by zero or taking the square root of a negative number.
• The range is the set of all possible outputs (y-values) that the machine can produce when you feed it valid inputs from the domain.

Understanding these concepts is crucial for analyzing functions, graphing them accurately, and applying them in real-world scenarios. The domain and range essentially define the boundaries within which our function operates, giving us a clear picture of its behavior. Mastering this foundational knowledge opens doors to more advanced topics in mathematics, making it a worthwhile investment of your time and effort.

Cracking the Domain of $y = 3\sqrt{x}$

Okay, now let's get our hands dirty with our function: $y = 3\sqrt{x}$. To find the domain, we need to ask ourselves: what values of x can we plug into this function without causing any mathematical mayhem? The big thing to watch out for here is the square root. Remember, you can't take the square root of a negative number (at least not in the realm of real numbers!). That would give us an imaginary number, and we're sticking with real numbers for this discussion.

So, the expression inside the square root, which is x in this case, must be greater than or equal to zero. Mathematically, we can write this as: $x \geq 0$. This is the key to unlocking the domain of our function. Any x-value that is zero or positive is fair game. We can plug it in, and the function will happily chug along, producing a real number output. But if we try to sneak in a negative value for x, the function will throw an error, signaling that we've stepped outside the allowed zone – the domain.

Therefore, the domain of the function $y = 3\sqrt{x}$ is all real numbers greater than or equal to zero. We can represent this in several ways: using inequality notation ($x \geq 0$), interval notation ($[0, \infty)$), or even graphically on a number line, where we'd shade the portion of the line from 0 to positive infinity, including 0 itself. Recognizing and defining the domain is a fundamental step in understanding the behavior and limitations of any function. It tells us the scope of inputs for which the function is valid and provides a crucial foundation for further analysis, such as graphing and problem-solving.

Unveiling the Range of $y = 3\sqrt{x}$

Alright, we've conquered the domain; now, let's tackle the range. To figure out the range, we need to think about all the possible y-values that our function can produce. We know that $y = 3\sqrt{x}$, and we've already established that $x \geq 0$ (our domain). So, let's consider what happens as x takes on values within this domain.

When x is 0, $y = 3\sqrt{0} = 0$. This gives us our starting point. As x increases from 0, the square root of x also increases. Since we're multiplying the square root of x by 3, the y-value will also increase. There's no upper limit to how large x can be (it can go to infinity), so the square root of x can also become infinitely large. Multiplying an infinitely large number by 3 still results in an infinitely large number.

This means that the y-values can be any non-negative number. We'll never get a negative y-value because the square root of a non-negative number is always non-negative, and multiplying a non-negative number by 3 keeps it non-negative. So, the range of the function $y = 3\sqrt{x}$ is all real numbers greater than or equal to zero. Just like the domain, we can represent this as $y \geq 0$, $[0, \infty)$, or graphically on a number line. Understanding the range is as important as understanding the domain. It completes the picture of how the function behaves, telling us the full spectrum of possible output values. By determining both the domain and range, we gain a comprehensive understanding of the function's capabilities and limitations.

The Grand Finale: Identifying the Correct Answer

Now that we've meticulously dissected the domain and range of $y = 3\sqrt{x}$, let's revisit the answer choices presented to us. We found that the domain is $x \geq 0$ (all non-negative real numbers) and the range is $y \geq 0$ (all non-negative real numbers). Looking at our options, we can confidently identify the correct answer:

• D. Domain: $x \geq 0$, Range: $y \geq 0$

This answer perfectly matches our findings. The other options are incorrect because they either restrict the domain to values less than or equal to zero, include all real numbers (which would allow for negative values inside the square root), or suggest a range that includes negative values (which is impossible for this function). By systematically working through the process of determining the domain and range, we not only arrived at the correct answer but also deepened our understanding of how functions behave.

Wrapping Up: Why Domain and Range Matter

So, there you have it, guys! We've successfully navigated the world of domains and ranges, specifically for the function $y = 3\sqrt{x}$. But why is this important? Understanding the domain and range of a function is like understanding the rules of a game. It tells you what moves are allowed and what outcomes are possible.

In the context of functions, the domain tells us what inputs we can use, and the range tells us what outputs we can expect. This knowledge is crucial for:

• Graphing functions accurately: Knowing the domain and range helps us set up the axes of our graph and plot the function correctly.
• Solving equations: Understanding the domain can help us identify extraneous solutions (solutions that don't actually work in the original equation).
• Modeling real-world situations: Many real-world phenomena can be modeled using functions, and understanding the domain and range helps us interpret the results in a meaningful way. For example, if we're modeling the height of a projectile, the domain might represent time (which can't be negative), and the range might represent the possible heights the projectile can reach.

By mastering the concepts of domain and range, you're equipping yourselves with essential tools for success in mathematics and beyond. So, keep practicing, keep exploring, and keep unraveling the mysteries of functions! Remember, every function has a story to tell, and the domain and range are key chapters in that story.