Finding the Inverse of a Function

In this lesson, we will explore the process of finding the inverse of a one-to-one function. To begin, we start with a given one-to-one function, often denoted as f(x), where each input value (x) corresponds to a unique output value (y) and vice versa. The key idea is to write the function as y=f(x) and then swap the roles of x and y in our equation. This step is crucial as it sets the foundation for finding the inverse function. By swapping x and y, our equation becomes x=f(y). Now, our task is to solve this equation for y in terms of x. This process involves applying algebraic techniques to isolate y on one side of the equation. Once we obtain y=..., we replace y with the notation f^-1(x) to represent the inverse function. It's essential to emphasize that the notation f^-1(x) is read as "f inverse of x" and not as "f to the power of -1 times x." The inverse function serves as a reversal of the original function, mapping values from the output (range) back to the input (domain). We often think about a function and its inverse as undoing each other. This relationship is often demonstrated by the composition of functions, where f(f^-1(x))=x holds true. The ability to find the inverse of a one-to-one function enables us to retrieve the original input value from the output, offering a powerful tool for analyzing and solving problems in various mathematical contexts.