Rewind the Step-Reverser

INVERSE FUNCTIONS — the rule that undoes another rule; to reverse a function you undo its steps in the opposite order, so an input sent through both comes back unchanged.

A story read by Rewind the Step-Reverser

Loading audio…

Press play to listen along. The line being read lights up as you go.

Show full transcript

Loading transcript…

01 Opening

Rewind was a mountain guide's apprentice.

Her master led travellers over the high passes — up through the pine woods, left at the split boulder, across the shale, right along the ridge, down into the far valley. It was Rewind's job, at the end of each crossing, to bring the party back. And the back way, she learned early, was not a new path. It was the same path — walked in reverse.

But there was a trick to it, and Rewind understood the trick before she understood that it was one. To reverse a journey, you did not simply flip the list of turns. You undid each step, in the opposite order. The last thing you had done was the first thing you undid. If the final leg had been down into the valley, the way back began up out of the valley. If the step before that had been right along the ridge, the next undoing was left along the ridge. Backwards through the list, and each turn reversed.

"You unwrap a journey the way you unwrap a parcel," she would say, coiling her rope. "Last knot tied is the first knot loosed."

02 Rewind the Step-Reverser

What Rewind eventually saw, guiding season after season, was that anything done in steps could be undone by reversing the steps in reverse order — and that when you did, you landed exactly where you started, not a pace off.

She tested it obsessively. She would walk a stranger blindfold up a winding trail, then bring them back by strict reversal, and set them down on the very stone they had begun from. Every time. The forward journey and the backward journey were not two different things. They were one path, and knowing it forward meant she could always, always find the way home.

She had no idea this had a name in the world of numbers.

03 Rewind the Step-Reverser

When she was older, a letter came from the FunctionForge academy, which was looking for someone who understood how a rule could be run backwards. Rewind, who had brought enough travellers home for one lifetime, coiled her rope, and went.

04 Rewind the Step-Reverser

At the academy she taught the children that every rule that does something can be undone.

"A function is a set of steps," she said, and wrote one on the board: f(x) = 2x + 3. "This one takes your number, doubles it, then adds three. Put in 5: double to 10, add three, 13. So f(5) = 13." The children nodded.

"Now — I am a guide. I bring it home." She wrote the journey backward. "The last step was add three, so the first undoing is subtract three. The step before was double, so the next undoing is halve. Backwards through the list, each step reversed." She wrote: f⁻¹(x) = (x − 3) ÷ 2.

"Watch it come home. Take our answer, 13. Subtract three: 10. Halve it: 5." She underlined the 5. "The number we started with. Sent forward through the rule and backward through its inverse — and it lands on the very stone it began from. Not a pace off."

"So the inverse just walks the same path backwards," said a girl.

"The same path, backwards, last step first," said Rewind. "That is the whole of an inverse. Undo the steps in the opposite order, and any number finds its way home."

05 Closing

She still guides a pass or two each summer, when the academy can spare her, and still cannot resist setting a traveller down on the exact stone they left from.

The thing Rewind hopes the children carry is that nothing careful is ever truly lost — that a set of steps taken can be a set of steps untaken, if you undo them in reverse, and that knowing the way back is what makes it safe to go forward at all.

"Every rule has a way home," she tells them, coiling an imaginary rope. "Reverse the steps, last one first, and you'll always land where you began."

When the hall empties, Rewind loops her old climbing rope over one shoulder and feels the steady, homeward gladness she first felt bringing travellers down out of the mountains: that no journey worth taking has to be a journey you can't come back from.

The FunctionForge ensemble

Rewind the Step-Reverser is part of FunctionForge's distributed-narrative cast. Each character embodies a different curricular primitive; together they teach the full subject.