Inverse Functions Explained: Step-by-Step Guide

Let's explore inverse functions and how they reverse mathematical operations.An inverse function undoes what the original function did. If we input x into f, then input the result into f inverse, we get back to x.For example, if our function doubles a number, its inverse will halve it.Let's see how these functions look on a graph. The blue line shows our original function, f of x equals two x.The red line shows its inverse function, f inverse of x equals x over two.Notice how these functions are reflections of each other across the line y equals x.Here are some specific examples of how inverse functions work.This special relationship means that when we compose a function with its inverse, in either order, we get back our input value.Let's follow a number through both functions. Starting with two, the original function doubles it to four, then the inverse function halves it back to two.Now that we understand what inverse functions are, let's see how to find them.To find an inverse function, we follow three main steps.Let's apply these steps to find the inverse of f of x equals two x plus three.First, we write our original function.Then we replace f of x with y to create an equation.Next, we swap x and y throughout the equation.Finally, we solve the equation for y to get our inverse function.To verify our answer, we check if f of f inverse of x equals x.Let's substitute f inverse of x into our original function.Multiply through by two.Simplify the fraction.And we get x, confirming that we found the correct inverse function.Now try finding the inverse of f of x equals three x minus four using these same steps.Not every function can have an inverse. To understand why, we need to look at a special property called one-to-one.A function is one-to-one if each y-value corresponds to exactly one x-value.We can use the horizontal line test to check this. If a horizontal line intersects the graph more than once, the function is not one-to-one.For functions like x squared that aren't one-to-one, we can restrict the domain to make them invertible.By restricting x squared to only non-negative x values, we get a function that is one-to-one. Its inverse is the square root function.