Verifying Inverse Functions F(x) = 5x - 25 And G(x) = (1/5)x + 5

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Hey guys! Let's dive into the fascinating world of inverse functions, where we'll explore how to verify if two functions are indeed inverses of each other. We'll be focusing on the specific functions f(x) = 5x - 25 and g(x) = (1/5)x + 5. This is a crucial concept in mathematics, and understanding it will empower you to solve a wide range of problems. So, let's get started!

Understanding Inverse Functions

Inverse functions are essentially functions that "undo" each other. Think of it like this: if you have a function that adds 5 to a number, its inverse would subtract 5 from that number. More formally, if f(x) is a function and g(x) is its inverse, then applying f and then g (or vice versa) will return you to your original input. This can be expressed mathematically as f(g(x)) = x and g(f(x)) = x. This is the core principle we'll use to verify if g(x) is the inverse of f(x). So, when we talk about inverse functions, we're really talking about a symmetrical relationship where each function reverses the operation of the other. Understanding this concept is fundamental not just for solving mathematical problems but also for grasping broader concepts in fields like computer science and engineering, where reversing processes is a common theme. Now, let's delve deeper into how we can apply this understanding to our specific functions.

The Key to Verification: Composition of Functions

To verify if g(x) is the inverse of f(x), we need to perform what's called the composition of functions. This means plugging one function into another. Specifically, we need to check two things: f(g(x)) and g(f(x)). If both of these compositions simplify to x, then we can confidently say that g(x) is indeed the inverse of f(x). So, the composition of functions isn't just a mathematical operation; it's a powerful tool for verifying relationships between functions, especially when dealing with inverses. This process allows us to see how one function "unravels" the operations performed by the other, bringing us back to the original input. It's like having a secret code and its decoder – when you apply both, you get back your original message. This concept is vital for anyone working with mathematical models, as it allows us to ensure that our models are reversible and consistent. Now, let's apply this concept to our specific functions and see how it works in practice.

Applying the Concept to f(x) and g(x)

In our case, we have f(x) = 5x - 25 and g(x) = (1/5)x + 5. Let's start by finding f(g(x)). This means we'll substitute g(x) into f(x) wherever we see x. So, we get f(g(x)) = 5((1/5)x + 5) - 25. Now, we need to simplify this expression. First, distribute the 5: 5 * (1/5)x = x and 5 * 5 = 25. So, we have f(g(x)) = x + 25 - 25. Simplifying further, we get f(g(x)) = x. Great! One condition is met. But we're not done yet. We also need to check g(f(x)). This involves substituting f(x) into g(x). This meticulous process of substitution and simplification is what allows us to definitively prove whether two functions are inverses. It's like a step-by-step verification process that leaves no room for doubt. By working through these steps, we gain not just the answer, but also a deeper understanding of the underlying mathematical principles. So, let's proceed with the second part of our verification process and see if g(f(x)) also simplifies to x.

Calculating g(f(x))

Now, let's calculate g(f(x)). This means we'll substitute f(x) = 5x - 25 into g(x) = (1/5)x + 5. So, we have g(f(x)) = (1/5)(5x - 25) + 5. Again, we need to simplify this expression. First, distribute the (1/5): (1/5) * 5x = x and (1/5) * -25 = -5. So, we get g(f(x)) = x - 5 + 5. Simplifying further, we find that g(f(x)) = x. Awesome! Both f(g(x)) and g(f(x)) simplify to x. This confirms that g(x) is indeed the inverse of f(x). This symmetrical relationship, where each function undoes the other, is the hallmark of inverse functions. The fact that both compositions simplify to x is not just a coincidence; it's a mathematical guarantee that the functions are inverses. This thorough verification process highlights the importance of precision and attention to detail in mathematics, ensuring that our conclusions are not just plausible, but mathematically sound.

Identifying the Correct Expression

Now that we understand the process, let's revisit the original question. We were asked which expression could be used to verify that g(x) is the inverse of f(x). Based on our work, we know that we need to check both f(g(x)) and g(f(x)). The expression for f(g(x)) is 5((1/5)x + 5) - 25, which simplifies to x. The expression for g(f(x)) is (1/5)(5x - 25) + 5, which also simplifies to x. Therefore, the correct expression to verify that g(x) is the inverse of f(x) is (1/5)(5x - 25) + 5, as this represents the composition g(f(x)). This process of identifying the correct expression highlights the importance of understanding the underlying mathematical concepts. It's not just about memorizing formulas; it's about understanding how those formulas are derived and applied. By grasping the concept of inverse functions and their verification through composition, we can confidently navigate through different expressions and identify the one that accurately represents the relationship between the functions.

Conclusion: Mastering Inverse Functions

So, guys, we've successfully navigated the world of inverse functions! We've learned what inverse functions are, how to verify them using composition, and applied this knowledge to the specific functions f(x) = 5x - 25 and g(x) = (1/5)x + 5. Remember, the key to verifying inverse functions is to check if f(g(x)) = x and g(f(x)) = x. By mastering this concept, you'll be well-equipped to tackle a wide range of mathematical problems involving inverse functions. Keep practicing, and you'll become a pro in no time! This understanding not only strengthens your mathematical foundation but also enhances your problem-solving skills in various other fields. The ability to reverse processes and understand symmetrical relationships is a valuable asset in any domain, be it engineering, computer science, or even everyday life. So, keep exploring, keep questioning, and keep mastering these fundamental concepts.