Finding The Value Of (f/g)(5) When F(x) = 7 + 4x And G(x) = 1/(2x)

by Sharif Sakr 67 views

Hey guys! Let's dive into a fun math problem where we'll explore functions, division, and how to find the value of a combined function at a specific point. We're given two functions, f(x) = 7 + 4x and g(x) = 1/(2x), and our mission is to find the value of (f/g)(5). Sounds like a plan? Let's get started!

Breaking Down the Problem: Understanding (f/g)(x)

Before we jump into plugging in numbers, let's understand what (f/g)(x) actually means. This notation represents the division of two functions. In simpler terms, it's like saying, "Hey, let's take the function f(x) and divide it by the function g(x)." So, (f/g)(x) is the same as f(x) / g(x). This understanding is crucial for solving the problem correctly.

Now, let's express this mathematically using the functions we have. We know that f(x) = 7 + 4x and g(x) = 1/(2x). Therefore, (f/g)(x) = (7 + 4x) / (1/(2x)). This looks a bit complex, right? But don't worry, we'll simplify it in the next step. Remember, the key here is to recognize that dividing by a fraction is the same as multiplying by its reciprocal. This is a fundamental concept in algebra, and it's going to be our best friend here.

To simplify, we'll multiply (7 + 4x) by the reciprocal of 1/(2x), which is 2x. So, we have (f/g)(x) = (7 + 4x) * (2x). Now, we need to distribute the 2x across the terms inside the parentheses. This means we'll multiply 2x by both 7 and 4x. When we do this, we get (f/g)(x) = 14x + 8x². See? It's already looking much simpler! This simplified expression for (f/g)(x) is what we'll use to find the value at x = 5. Remember, understanding the basic principles of function division and simplification is essential for tackling these kinds of problems. It's like having the right tools for the job – it makes everything easier and more efficient.

Calculating (f/g)(5): Plugging in the Value

Now that we have a simplified expression for (f/g)(x), which is 14x + 8x², the next step is to find the value of this expression when x = 5. This means we're going to substitute 5 for every x in the expression. This is a common technique in algebra and calculus, and it's how we evaluate functions at specific points. Think of it like this: the function is a machine, and we're feeding it the number 5 as input. The machine will then process this input and give us an output, which is the value of the function at that point.

So, let's plug in x = 5 into our expression: (f/g)(5) = 14(5) + 8(5)². Now, we need to follow the order of operations (PEMDAS/BODMAS), which tells us to do the exponent first, then multiplication, and finally addition. This is crucial for getting the correct answer. First, we calculate 5², which is 5 * 5 = 25. So, our expression becomes (f/g)(5) = 14(5) + 8(25). Next, we perform the multiplications: 14(5) = 70 and 8(25) = 200. Now, our expression is (f/g)(5) = 70 + 200. Finally, we add the two numbers together: 70 + 200 = 270. Therefore, (f/g)(5) = 270. Isn't it satisfying when you arrive at the final answer? Remember, the key here was to carefully substitute the value and follow the order of operations.

The Grand Finale: The Value of (f/g)(5)

We've journeyed through the world of functions, division, and substitution, and now we've arrived at our destination: the value of (f/g)(5). As we meticulously calculated, by first understanding that (f/g)(x) is simply f(x) divided by g(x), then simplifying the expression, and finally plugging in x = 5, we found that (f/g)(5) = 270. This result represents the value of the combined function f/g when evaluated at the point x = 5. This is the culmination of all our hard work, and it's something to be proud of!

So, the final answer to our problem is 270. This number tells us something specific about the relationship between the functions f(x) and g(x) at the point x = 5. It's a single, concrete value that encapsulates the combined behavior of these two functions. Understanding how to find such values is a fundamental skill in mathematics, and it opens the door to more advanced concepts in calculus and analysis. So, give yourself a pat on the back for conquering this problem! And remember, the journey of mathematical discovery is a continuous one, with each problem solved paving the way for new challenges and new insights.

Key Takeaways and Further Exploration

Throughout this problem, we've encountered several important mathematical concepts that are worth highlighting. These concepts are the building blocks for more advanced topics, and understanding them thoroughly will be beneficial in your future mathematical endeavors.

  • Function Division: We learned that (f/g)(x) means dividing the function f(x) by the function g(x). This is a fundamental operation on functions, and it's used extensively in calculus and analysis.
  • Simplifying Expressions: We simplified the expression for (f/g)(x) by using the principle that dividing by a fraction is the same as multiplying by its reciprocal. This is a crucial skill in algebra, and it's essential for making complex expressions more manageable.
  • Substitution: We substituted x = 5 into the simplified expression to find the value of (f/g)(5). This technique is used throughout mathematics to evaluate functions and solve equations.
  • Order of Operations: We followed the order of operations (PEMDAS/BODMAS) to ensure that we performed the calculations in the correct sequence. This is essential for getting the correct answer in any mathematical problem.

To further explore these concepts, you might want to try similar problems with different functions or different values of x. You could also investigate other operations on functions, such as addition, subtraction, and composition. The more you practice, the more comfortable you'll become with these concepts, and the more confident you'll feel in your mathematical abilities. So, keep exploring, keep practicing, and keep having fun with math!

This was a great exercise in understanding functions and how they interact with each other. Keep up the awesome work, guys! You're doing great!