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Precalculus 9-12 / Week 04 / Friday
5/6
Week 04 · Inverses Everywhere

Friday

Inverses around the Boathouse
// The winch rule run backwards
⏱ about 20 min

Friday: Inverses Around the Boathouse

Friday morning the Boathouse wall is covered in rules and their inverses from the week.

"10 turns for 6 meters. Session 5 for a 5 minute lap. 4 clicks for a reading of 48," Comet reads.

"Three different rules," Wren says. "What do you notice they share?"

"Each one got run backwards," Comet says. "Output in, input out."

Nova sorts the sheets into two columns. "Would you like a hint? One column is rules, the other is their mirrors."

"The same pairs, read both ways," Wren says. "Today we read both columns without slowing down."

"And then I want one with base 10," Comet says. "Nova's tally grid counts in tens."

Where the crew used inverses this week

QuestionRuleInverseAnswer
Turns for 6 meters of ropef(x) = 0.5x + 1f⁻¹(x) = 2x - 210 turns
Session for a 5 minute lapL(x) = (3x + 20)/(x + 2)L⁻¹(x) = (-2x + 20)/(x - 3)session 5
Clicks for a glow reading of 48g(x) = 3 × 2ˣg⁻¹(x) = log₂(x/3)4 clicks
Turns for 70 cm on the lab tubef(x) = 10x + 30f⁻¹(x) = 0.1x - 34 turns

Every row has the same shape: a rule, its inverse, and a backwards question the inverse answers at once.

When the rule is a line, undo the steps. When it is a fraction, swap and solve. When it is exponential, take a logarithm.

Base 10 in the tally

Nova's tally grid counts in tens: 1, 10, 100, 1,000. The rule h(x) = 10ˣ has the inverse h⁻¹(x) = log₁₀ x.

log₁₀ 1,000 = 3 because 10³ = 1,000. Base 10 logarithms count the zeros on a power of ten.

Any exponential rule has a logarithm as its inverse. The base of the log is the base of the power.

Which answer makes sense?

Solving the winch rule backwards for 6 meters gives 10 turns. Solving it for 0.5 meter gives -1 turns.

A negative turn count is true algebra and still outside the log the crew kept. Say so when it happens.

MIXED SET
  • Read the question.
  • Tap your answer.
A tally starts at 5 and multiplies by 10 each step: h(x) = 5 × 10^x. What is its inverse?
The tally grid: log₁₀ 1,000 = ?
A rule in the log reads f(x) = (x + 2)/(x - 3). What is its inverse?
f(x) = 0.5x + 1 and f⁻¹(x) = 2x - 2. What is f(f⁻¹(6))?
MIXED SET, PART TWO
  • Read the question.
  • Tap your answer.
The dimmer, from a different start: 2^x = 64. Solve for x.
A tally sentence reads log₁₀(x) = 4. What is x?
The winch table: f(0) = 1, f(2) = 2, f(4) = 3, f(6) = 4, f(8) = 5, f(10) = 6. What is f⁻¹(3)?
f(x) = 0.5x + 1. Let g(x) = 2f(x). What is g(4)?
log₂ 32 = ? Type the number.
log₁₀ 10,000 = ? Type the number.
The lap-time inverse is L⁻¹(x) = (-2x + 20)/(x - 3). Which session gives a 4.4 minute lap? Type the number.
StatementTrue or false?
The inverse of y = 10ˣ is y = log₁₀ x.?
10 × 10 × 10 = 1,000?
f(f⁻¹(x)) = x for every x the inverse accepts.?
The inverse of a one-to-one function can send one input to two outputs.?
Shifting a rule up 3 shifts its inverse right 3.?
WHY THIS EXERCISEThe week in five lines: swap, solve, compose to check, read the mirror, and let a log undo a power.
Draw the four rows of the table as four pairs of arrows: forwards with the rule, backwards with the inverse.

A full week of rules run backwards. Tomorrow is Dock Day, and a rule from your own home gets its inverse.

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