How to inverse a function in a set

Let’s take an example of a magic machine that takes a toy and turns it into a box. Now, we want to get our toy back from the box. For that, we need another machine that would be the inverse–it takes the box and gives us back our toy.

Similarly, in maths, when we have a function that turns one thing into another (like turning a number into its square), the inverse function does the opposite. It takes the result and gives us back the original number.

Bijective function

If a function is one-to-oneEach element in the domain maps to a unique element in the codomain. and ontoThe function covers every element in the codomain; there are no leftovers. then we can call it a bijective function. When a function is bijective, it’s good for doing the reverse process, which we call finding the inverse.

Example of a bijective function

Consider the domain set $A=\{1, 2, 3\},$ the codomain set $B=\{1, 2, 3\}$ and the function $f(x) = x$. Its diagram is shown below.

Inverse function

An inverse function is like a magical undo button for a function. Simply put, when a function is bijective the input and output switch places. It’s like a special machine where the place we put things in becomes the place where things come out, and vice versa.

Let’s look at the same example we talked about earlier and try using the undo rule on the function:

We might notice that we flipped all the arrows in the special matching machine we discussed earlier (bijective function). This is how we create the backward, or undo, function, which we call the inverse function.

Let’s solve the quiz below to assess yourself: