Exploring The Function G(x) = (2x + 8) / X Calculation And Range Analysis

Hey guys! Today, we're diving deep into the fascinating world of functions, specifically focusing on the function g(x) = (2x + 8) / x. We'll break down how to evaluate this function for a given input and determine its range over a specific domain. So, grab your thinking caps, and let's get started!

Understanding the Function g(x)

Before we jump into the calculations, let's first understand what this function g(x) = (2x + 8) / x actually represents. In simple terms, a function is like a machine that takes an input, performs some operations on it, and produces an output. In this case, our function g(x) takes an input x, multiplies it by 2, adds 8 to the result, and then divides the entire sum by x. The result of this operation is the output, which we denote as g(x). Functions are a fundamental concept in mathematics, and they're used to model relationships between different quantities. Understanding functions is crucial for various areas of math and science, including calculus, algebra, and physics.

This particular function, g(x) = (2x + 8) / x, is a rational function, meaning it's a ratio of two polynomials. Rational functions have some interesting properties, such as potential vertical asymptotes (where the denominator equals zero) and horizontal asymptotes (which describe the function's behavior as x approaches infinity or negative infinity). In the context of this function, x cannot be zero because division by zero is undefined. This is a critical point to remember when working with rational functions. We also need to be mindful of the domain, which is the set of all possible input values for x, and the range, which is the set of all possible output values for g(x). When dealing with functions, especially rational functions, it's important to consider these constraints to avoid mathematical errors and to ensure the function's validity.

Now, why is it important to understand functions, especially this kind of function? Well, many real-world scenarios can be modeled using functions. Think about the relationship between the number of hours you work and the amount of money you earn, or the relationship between the distance a car travels and the amount of fuel it consumes. Functions help us make predictions, analyze trends, and solve problems in various fields. By mastering the basics of functions, you're equipping yourself with a powerful tool for tackling mathematical and real-world challenges. So, with this understanding in mind, let's move on to the first part of our problem: finding g(1/2).

(a) Finding g(1/2)

Okay, so the first part of our problem asks us to find g(1/2). What does this mean? It simply means we need to substitute x with 1/2 in our function g(x) = (2x + 8) / x. Let's walk through the steps together. First, we replace every instance of x in the function with 1/2. This gives us g(1/2) = (2(1/2) + 8) / (1/2). Now, we need to simplify this expression. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Following the order of operations, we first deal with the multiplication inside the parentheses: 2 multiplied by 1/2 equals 1. So, our expression becomes g(1/2) = (1 + 8) / (1/2). Next, we perform the addition in the numerator: 1 plus 8 equals 9. Now we have g(1/2) = 9 / (1/2). We're left with a fraction divided by another fraction. To divide by a fraction, we multiply by its reciprocal. The reciprocal of 1/2 is 2/1, which is simply 2. Therefore, g(1/2) = 9 * 2. Finally, we perform the multiplication: 9 multiplied by 2 equals 18. So, we've found that g(1/2) = 18. Woohoo! We've successfully evaluated the function for x = 1/2.

Now, let's think about what this result tells us. It tells us that when the input to our function g(x) is 1/2, the output is 18. This is a specific point on the graph of the function. If we were to graph g(x), the point (1/2, 18) would lie on the curve. Understanding how to evaluate functions for specific inputs is crucial for graphing functions, solving equations, and analyzing mathematical models. By practicing these calculations, you'll become more confident in your ability to work with functions and interpret their behavior. With this first part conquered, let's move on to the next challenge: finding the range of g under the domain {1, 2, 3, 4}.

(b) Finding the Range of g under the Domain {1, 2, 3, 4}

Alright, guys, let's tackle the second part of our problem: finding the range of the function g(x) = (2x + 8) / x under the domain {1, 2, 3, 4}. Now, what does this mean exactly? The domain is the set of all possible input values for x, and in this case, it's limited to the numbers 1, 2, 3, and 4. The range, on the other hand, is the set of all possible output values for g(x) when we use the values from the domain as inputs. So, basically, we need to plug in each value from the domain into the function and see what outputs we get. These outputs will form the range of g(x) under this specific domain.

Let's start by finding g(1). We substitute x with 1 in our function: g(1) = (2(1) + 8) / 1. Simplifying, we get g(1) = (2 + 8) / 1 = 10 / 1 = 10. So, the first output value is 10. Next, let's find g(2). We substitute x with 2: g(2) = (2(2) + 8) / 2. Simplifying, we get g(2) = (4 + 8) / 2 = 12 / 2 = 6. Our second output value is 6. Moving on, let's find g(3). We substitute x with 3: g(3) = (2(3) + 8) / 3. Simplifying, we get g(3) = (6 + 8) / 3 = 14 / 3. This is approximately 4.67. Finally, let's find g(4). We substitute x with 4: g(4) = (2(4) + 8) / 4. Simplifying, we get g(4) = (8 + 8) / 4 = 16 / 4 = 4. Our last output value is 4.

Now that we've calculated the output values for each input value in the domain, we can write the range of g(x) under the domain {1, 2, 3, 4}. The range is the set of these output values, which are {10, 6, 14/3, 4}. We can write this as a set of numbers, usually in ascending order, as {4, 14/3, 6, 10}. Remember, the range represents the possible output values of the function for the given domain. In this case, when we input 1, 2, 3, or 4 into the function g(x), the possible outputs are 4, 14/3, 6, and 10. Understanding how to find the range of a function for a given domain is essential for analyzing the function's behavior and its possible outputs within a specific context.

By finding the range, we've essentially mapped the inputs from our domain to their corresponding outputs in the range. This mapping gives us a clearer picture of how the function transforms the input values. It's like seeing the function in action, turning one set of numbers into another. This skill is super useful in many mathematical applications, from solving equations to creating models that predict real-world outcomes. So, give yourselves a pat on the back for mastering this concept! And now, let's wrap up our discussion with some final thoughts.

Conclusion

So there you have it, guys! We've successfully navigated the function g(x) = (2x + 8) / x. We learned how to evaluate the function for a specific input, like finding g(1/2), and how to determine the range of the function under a given domain, like {1, 2, 3, 4}. These are fundamental skills in mathematics, and mastering them will set you up for success in more advanced topics. Remember, functions are the building blocks of many mathematical models, and understanding how they work is crucial for solving real-world problems.

We saw how to substitute values into the function, simplify expressions using the order of operations, and identify the possible output values for a given set of inputs. We also emphasized the importance of considering the domain and range when working with functions, especially rational functions. The domain tells us which inputs are valid, while the range tells us the possible outputs we can expect. This understanding is key to avoiding errors and interpreting the function's behavior correctly.

Keep practicing these concepts, guys, and you'll become function whizzes in no time! Exploring functions is like unlocking a secret code to the mathematical universe. The more you practice, the more patterns you'll see, and the more confident you'll become in your problem-solving abilities. So, keep exploring, keep questioning, and keep learning! You've got this!