Unveiling Function Values & Domains: A Mathematical Journey

Hey Plastik Magazine readers! Let's dive into the fascinating world of functions, specifically focusing on how to determine their values and, importantly, their domains. Today, we'll explore a particular function, $f(x) = \sqrt{0.5x - 3}$, and calculate some specific values while also uncovering the secrets of its domain. Get ready to flex those mathematical muscles, because we're about to embark on an exciting problem-solving adventure together! This is going to be a fun journey, so stick around!

Understanding the Function and Our Goal

So, what exactly is a function, and what are we trying to accomplish here? In simple terms, a function is like a mathematical machine. You feed it an input (usually represented by 'x'), and it spits out an output (represented by 'f(x)'). The function itself dictates the rules of this transformation. In our case, the function is $f(x) = \sqrt{0.5x - 3}$. This means we take an input 'x', multiply it by 0.5, subtract 3, and then take the square root of the result. Our primary goals here are twofold: first, to calculate the output 'f(x)' for given input values, and second, to identify the domain of the function. The domain of a function is the set of all possible input values (x-values) for which the function produces a valid, real-number output. Understanding the domain is crucial because some inputs might lead to undefined results, especially with square roots. We need to be super careful!

We're going to calculate f(6), f(12), f(10), and f(8). Each of these is a different input, and we will do our best to find their results! Finding these values is like figuring out the result of our function when these numbers are put into the function! It can be a little tricky sometimes, but it’s definitely doable! Just think of it as plugging in different values into the equation and seeing what happens. We have to be aware of certain constraints on the square root and where it can take us, but we will always be careful about this. Let's get started, and I'll walk you through each step, making sure you understand everything! Let's make sure we do the math correctly and then show you the results! These are essential pieces to the whole picture, so we must be certain about them. This is going to be a fun ride for sure!

Calculating Function Values: Step-by-Step

Alright, let's roll up our sleeves and calculate those function values! We'll go through each one systematically, making sure to show every step, so it's super clear. Remember, we're simply substituting the given 'x' value into the function and simplifying. Don't worry, it's not as scary as it sounds. We'll start with f(6).

Calculating f(6)

To find f(6), we replace 'x' with 6 in our function: $f(6) = \sqrt{0.5(6) - 3}$. Now, let's simplify step-by-step. First, calculate 0.5 multiplied by 6, which equals 3. Our equation then becomes: $f(6) = \sqrt{3 - 3}$. Subtracting 3 from 3 gives us 0: $f(6) = \sqrt{0}$. Finally, the square root of 0 is 0. Therefore, $f(6) = 0$. Pretty easy, right?

Calculating f(12)

Next up, we have f(12). Following the same process, we substitute 'x' with 12: $f(12) = \sqrt{0.5(12) - 3}$. Multiplying 0.5 by 12 gives us 6: $f(12) = \sqrt{6 - 3}$. Subtracting 3 from 6 results in 3: $f(12) = \sqrt{3}$. The square root of 3 is approximately 1.732 (we can use a calculator for this, or leave it in radical form). Therefore, $f(12) = \sqrt{3}$ or approximately 1.732.

Calculating f(10)

Let's calculate f(10). Substitute 'x' with 10: $f(10) = \sqrt{0.5(10) - 3}$. Then, 0.5 times 10 is 5: $f(10) = \sqrt{5 - 3}$. Subtracting 3 from 5 yields 2: $f(10) = \sqrt{2}$. The square root of 2 is approximately 1.414. So, $f(10) = \sqrt{2}$ or approximately 1.414.

Calculating f(8)

Finally, let's find f(8). Substitute 'x' with 8: $f(8) = \sqrt{0.5(8) - 3}$. Then, 0.5 times 8 is 4: $f(8) = \sqrt{4 - 3}$. Subtracting 3 from 4 gives 1: $f(8) = \sqrt{1}$. The square root of 1 is 1. Thus, $f(8) = 1$. Great job, everyone! We've successfully calculated all the requested function values!

Unveiling the Domain of the Function

Now, let's shift gears and focus on the domain of our function, $f(x) = \sqrt{0.5x - 3}$. As mentioned earlier, the domain is the set of all possible input values (x-values) that produce a valid output. Here's where we have to be a little more careful, because we are dealing with a square root function. The key to determining the domain of a square root function is understanding that the expression inside the square root (the radicand) cannot be negative. Why? Because the square root of a negative number is not a real number; it's an imaginary number, and we're sticking to the real number system for this exercise. The value inside the square root must always be positive or equal to zero. If this rule is not followed, we run into some issues and the numbers don't work! We have to be aware of what is possible and what is not. Let's make sure our radicand stays at a good value.

So, for our function, $f(x) = \sqrt{0.5x - 3}$, we need to ensure that $0.5x - 3 \geq 0$. This inequality is what we'll use to find the domain. The equation is straightforward, but it's important to understand the concept of what it means. We are looking for the x-values that will make the function