Inverse functions

You wrap a present: first the paper, then the ribbon. To unwrap it you do everything backwards: first untie the ribbon, then remove the paper. That is exactly the idea behind inverse functions. A function turns an input into an output. Its inverse function undoes this: it turns the output back into the original input.

Running the machine backwards

Take the function $y = 2x + 3$. Read as a machine, it does two steps: first multiply by 2, then add 3. The input 4 becomes the output 11. To calculate backwards, you reverse both things: the order of the steps and each step itself. Backwards this means: first subtract 3, then divide by 2. 11 becomes 4 again. Just like with the present: what was done last is undone first.

Swap the table, mirror the graph

In a table of values, inverting is very simple: you swap the two columns. If the function has the pair input 2 and output 7, then the inverse function has input 7 and output 2. Something beautiful happens to the graph: because every point (a|b) becomes the point (b|a), the graph of the inverse function appears by mirroring at the line $y = x$. This line is the diagonal on which input and output are equal.

Not every machine can be reversed this easily. For linear functions like $y = 2x + 3$ it always works, as long as the slope is not 0. For $y = x^2$, on the other hand, both 3 and −3 give the output 9, so from the output alone the input can no longer be recovered uniquely. It gets exciting with functions like $y = 10^x$: it has an inverse too. That inverse answers the question of which exponent 10 has to be raised to in order to give a particular number. For the output 1000 that exponent is 3. This inverse is called the logarithm, and you will meet it in more detail later.