Finding The Inverse Of The Function F(x) = 2x - 10

Determining the inverse of a function is a fundamental concept in mathematics, particularly in algebra and calculus. The inverse function, denoted as f⁻¹(x), essentially "undoes" the operation performed by the original function, f(x). In simpler terms, if f(a) = b, then f⁻¹(b) = a. This article will walk you through the process of finding the inverse of the function f(x) = 2x - 10, providing a clear, step-by-step explanation to help you understand the underlying principles. We'll break down each stage, from the initial function definition to the final verification, ensuring that you grasp not only the method but also the logic behind it. By understanding how to find inverse functions, you'll be better equipped to tackle more complex mathematical problems and applications in various fields. So, let's embark on this journey to unravel the mystery of inverse functions and gain a solid understanding of this essential mathematical concept. Our goal is to make the process as straightforward and accessible as possible, ensuring that you can confidently apply these techniques to other functions in the future. The ability to find inverse functions is a valuable skill that opens doors to further exploration in mathematics and related disciplines.

Step 1: Replace f(x) with y

The first step in finding the inverse of the function is to replace the function notation, f(x), with the variable y. This substitution makes the equation easier to manipulate algebraically. So, in our case, we replace f(x) = 2x - 10 with y = 2x - 10. This simple change sets the stage for the subsequent steps and helps to visualize the function in a more traditional equation form. By using 'y' instead of 'f(x)', we can more readily apply algebraic techniques to isolate the variable and ultimately find the inverse function. This step is crucial because it transforms the function notation into a format that is more amenable to algebraic manipulation. It's a foundational step that simplifies the process of finding the inverse. Think of it as translating the function from one language to another, making it easier to work with. This change allows us to treat the function as a standard equation, which we can then manipulate to solve for the inverse. So, remember, the first step is always to replace f(x) with y, setting the stage for the next steps in our journey to find the inverse function.

Step 2: Swap x and y

Next, we perform a crucial step in the inverse function process: we swap the variables x and y. This is the heart of finding the inverse, as it reflects the function across the line y = x. By interchanging x and y, we are essentially reversing the roles of the input and output, which is the fundamental idea behind an inverse function. So, our equation y = 2x - 10 becomes x = 2y - 10. This swap is not just a mechanical step; it embodies the very definition of an inverse function. The new equation represents the inverse relationship, where the original output (y) now becomes the input (x), and vice versa. This step is vital because it sets up the equation in a form where we can solve for the new 'y', which will represent the inverse function. It's like looking at the function from a reverse perspective, where the dependent and independent variables exchange their roles. This swapping of variables is a key maneuver that allows us to unravel the original function and reveal its inverse. It's a fundamental operation that directly leads us to the solution.

Step 3: Solve for y

Now, we need to isolate y on one side of the equation. This involves using algebraic manipulation to solve for y in terms of x. Starting with our swapped equation, x = 2y - 10, we'll perform a series of operations to get y by itself. First, we add 10 to both sides of the equation: x + 10 = 2y. Then, we divide both sides by 2 to isolate y: (x + 10) / 2 = y. This process of solving for y is a core skill in algebra and is essential for finding the inverse function. Each step we take is aimed at unraveling the equation and revealing the relationship between y and x in the inverse form. By carefully applying algebraic principles, we can isolate y and express it as a function of x. This step requires attention to detail and a solid understanding of algebraic manipulation. It's like piecing together a puzzle, where each operation brings us closer to the final solution. The goal is to express y explicitly in terms of x, which will give us the equation for the inverse function. So, by systematically isolating y, we are uncovering the inverse relationship and preparing to express it in standard function notation.

Step 4: Rewrite y as f⁻¹(x)

Having solved for y, the final step is to rewrite y using inverse function notation. We replace y with f⁻¹(x), which represents the inverse function of f(x). From our previous step, we have y = (x + 10) / 2. Now, we rewrite this as f⁻¹(x) = (x + 10) / 2. This notation clearly indicates that we have found the inverse function. It's a standard way of representing inverse functions and is universally understood in mathematics. The notation f⁻¹(x) is a concise and elegant way of expressing the inverse relationship. It signifies that this new function undoes the operation of the original function, f(x). This step is not just about notation; it's about formally declaring that we have successfully found the inverse function. It's like putting the final stamp on our work, indicating that we have completed the process. The inverse function, f⁻¹(x) = (x + 10) / 2, is the function that, when composed with f(x), will result in the identity function, x. So, by rewriting y as f⁻¹(x), we are formally presenting the inverse function and making it clear that we have achieved our goal.

To further simplify and match the answer options, we can distribute the division: f⁻¹(x) = x/2 + 10/2, which simplifies to f⁻¹(x) = (1/2)x + 5.

Solution

Therefore, the inverse of the function f(x) = 2x - 10 is h(x) = (1/2)x + 5.

Final Answer: D. h(x) = (1/2)x + 5

In conclusion, finding the inverse of a function involves a systematic approach. We started by replacing f(x) with y, then swapped x and y, solved for y, and finally rewrote y as f⁻¹(x). This process allowed us to determine that the inverse of f(x) = 2x - 10 is indeed h(x) = (1/2)x + 5. Understanding this process is crucial for further mathematical studies and applications. The ability to find inverse functions is a valuable tool in various fields, including calculus, algebra, and beyond. By mastering these steps, you'll be well-equipped to tackle more complex mathematical problems and gain a deeper understanding of functional relationships. The systematic approach we've outlined ensures that you can confidently find the inverse of a wide range of functions. It's a skill that builds upon fundamental algebraic principles and opens doors to more advanced mathematical concepts. So, practice these steps, and you'll become proficient in finding inverse functions, a skill that will serve you well in your mathematical journey.