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Precalculus 9-12 / Week 04 / Dock Day
6/6
Week 04 · Inverses Everywhere

Dock Day

Dock Day: a home rule backwards
// The winch rule run backwards
⏱ about 20 min

Dock Day: A Home Rule Backwards

Saturday the crew sends everyone home with one job: find a rule and run it backwards.

"A staircase is a rule," Comet says. "Height climbed equals the riser height times the steps."

"A tape measure is a rule," Wren adds. "Centimeters in, inches out. Or the other way."

Nova dims to a soft glow. "Would you like a hint? Measure first, then write the rule with x and y."

"Then swap, solve, and check one pair," Comet says. "Show the family both directions."

"And draw both on one grid with the mirror line," Wren says. "The fold is the proof."

  1. An inverse undoes a rule: if f(a) = b, then f⁻¹(b) = a. To find it, swap x and y and solve, or undo each step last step first.
  2. Only a one-to-one function has an inverse function. Two inputs sharing an output would swap into one input with two outputs.
  3. Check an inverse by composing both ways: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
  4. A logarithm is the inverse of an exponential: log₂ 16 = 4 because 2⁴ = 16. Divide away the start, then take the log.
  5. The inverse graph is the mirror across y = x. Up on one side is right on the other.
◇ FAMILY RULE, RUN BACKWARDS
Choose a home rule together: a staircase, a measuring tape, a bike pump's strokes, or a rope wound on a spool.
Measure three pairs and write the rule with x and y. Someone else writes the inverse by swapping and solving.
Pick a target output. Use the inverse to predict the input, then measure to check.
Plot the rule and its inverse on one sheet of graph paper. Fold along y = x. Trade roles and try a second rule.
For a grown-up
This is a measuring activity with a tape measure and graph paper. Stairs are measured standing still, one hand on the rail.
A wrong prediction is a good moment: check which step of the inverse was undone in the wrong order.
The rules are the family's own measurements, not facts about any real staircase or tape.
DOCK DAY CHECK
  • Read the question.
  • Tap your answer.
A staircase table in centimeters: f(1) = 20, f(2) = 40, f(3) = 60, f(4) = 80. What is f⁻¹(60)?
A porch step of 10 cm then risers of 20 cm: height h(x) = 20x + 10 after x stairs. What is the inverse?
A spool doubles its string each wrap, from 1 to 16. How many wraps is that? That is log₂ 16.
The winch rule f(x) = 0.5x + 1 and its inverse 2x - 2. What is f⁻¹(f(12))? Type the number.
WHY THIS EXERCISEComposing both ways and getting x back is the one test every inverse must pass.
Draw your home rule and its inverse on one grid with the dashed line y = x.
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