Inverse Functions Explained with Simple Animations

Let's explore inverse functions and understand how they undo what the original function does.We'll start with a simple function: f of x equals two x plus one.Let's see what happens when we input the number two into this function.When we put two into our function, first we multiply it by two, getting four, then add one, getting five.Now, the inverse function undoes what f of x did. It's written as f inverse of x, and equals x minus one, all divided by two.Watch how the inverse function takes our output of five, and returns us to our original input of two.Think of these functions as machines. The original function takes our input and transforms it, while the inverse function reverses that transformation.When we put two into our function machine, it becomes five, but then the inverse function machine turns it back into two.Now let's explore how inverse functions are related graphically.First, let's plot our original function f of x equals two x plus one in blue.The line y equals x plays a crucial role in understanding inverse functions.Let's see how points from our original function reflect across the line y equals x to create the inverse function.As we connect these reflected points, we get the inverse function f inverse of x equals x minus one over two, shown in red.Notice how each point's coordinates are swapped when reflected. The x coordinate becomes the y coordinate, and vice versa.These reflections show us important properties of inverse functions: they reflect across y equals x, their coordinates swap positions, and their slopes are reciprocals of each other.To find the inverse of f of x equals two x plus one, we'll follow a step-by-step process.First, we replace f of x with y.Next, we swap x and y.Now we solve for y. First subtract one from both sides, then divide both sides by two.Finally, we replace y with f inverse of x.Let's verify this is truly the inverse function by composing f of f inverse of x.And also f inverse of f of x.Let's see how these functions work together with an example. Starting with five.First, f of five equals eleven.Then, f inverse of eleven equals five, returning to our original input.Similarly, starting with seven.f inverse of seven equals three.And f of three equals seven, again returning to our starting value.