Monday
Monday: Run the Rule Backwards
Rain has left the ramp slick, so the rowboat waits on its cradle while the crew checks the winch.
"Nova's log says the drum holds 1 meter of rope before the first turn," Wren says. "Each turn winds on 0.5 meter more."
"So rope on the drum is 0.5 times turns, plus 1," Comet says. "What can we make of that?"
"Run it backwards," Wren says. "The boat needs 6 meters hauled in. How many turns is that?"
Nova hovers over the drum. "Would you like a hint? Undo the rule one step at a time, last step first."
"Subtract the 1, then double," Comet says. "6 minus 1 is 5, doubled is 10. 10 turns."
"That backwards rule is called the inverse," Wren says. "This week every rule in the log gets run backwards."
A rule and its inverse
The winch rule is f(x) = 0.5x + 1: put in turns, get out meters of rope on the drum.
Its inverse is f⁻¹(x) = 2x - 2: put in meters of rope, get back the turns. Read f⁻¹ as "f inverse."
Check with one pair. f(10) = 6, and f⁻¹(6) = 10. Each sends the other's output back home.
The inverse is a new function, not a reciprocal. f⁻¹(x) never means 1 divided by f(x).
A solved problem to study
- Write the rule with y: y = 0.5x + 1.
- Swap x and y, because the inverse swaps inputs and outputs: x = 0.5y + 1.
- Solve for y. Subtract 1: x - 1 = 0.5y.
- Divide by 0.5, which is the same as doubling: y = 2x - 2.
- Name it: f⁻¹(x) = 2x - 2.
- Check: f(4) = 3 and f⁻¹(3) = 4. It undoes.
Notice the order inside the inverse. The rule adds last, so the inverse subtracts first. The rule halves first, so the inverse doubles last.
Every number here is the crew's own winch reading from Nova's log, not a fact about real winches.
The winch table, forwards and backwards
| Turns (x) | Rope on the drum (m) | Read backwards |
|---|---|---|
| 0 | 1 | f⁻¹(1) = 0 |
| 2 | 2 | f⁻¹(2) = 2 |
| 4 | 3 | f⁻¹(3) = 4 |
| 6 | 4 | f⁻¹(4) = 6 |
| 8 | 5 | f⁻¹(5) = 8 |
| 10 | 6 | f⁻¹(6) = 10 |
- Read the question.
- Tap your answer.
| Statement | True or false? |
|---|---|
| f(x) = 0.5x + 1 has the inverse f⁻¹(x) = 2x - 2. | ? |
| f⁻¹(x) means 1 divided by f(x). | ? |
| 0.5 × 10 + 1 = 6 | ? |
| To find an inverse, swap x and y, then solve for y. | ? |
| A rule where two inputs share one output still has an inverse function. | ? |
On the back, write the undoing steps in reverse order. Test both with the number 10.
Strong start. Tomorrow the lap-time rule and the dimmer rule get run backwards, and a logarithm appears.