← Back to course
Algebra 2 9-12 / Week 05 / Garden Day
6/6
Week 05 · Radicals and Inverse Functions

Garden Day

Garden Day: the backwards machine
// The drip hose, forwards and backwards
⏱ about 20 min

Garden Day: The Backwards Machine

Saturday the index cards come home, and the kitchen table becomes a rule machine with a reverse gear.

"Each card has a rule," Comet says. "You run my number through it. Then I give you an output and you run it backwards."

The first card reads "times 4, then add 10." Someone puts in 5 and calls out "Thirty!"

"Now backwards from 50," Wren says. "Ten," comes the answer. "Subtract 10, divide by 4."

Nova hovers above the cards. "Would you like a hint? Write the backwards rule as a function of its own."

"x minus 10, over 4," Comet says. "Our drip rule and its inverse, on a kitchen table."

"Next card is "square it"," Wren says. "Backwards from 25. How many answers? That is today's trick question."

  1. A radical equation: square both sides, solve, then check. Squaring can create an extraneous solution.
  2. A rational exponent is a root on the bottom and a power on top: 8^(2/3) = 4.
  3. An inverse function runs a rule backwards. Build it by swapping x and y and solving for y.
  4. Only a one-to-one function has an inverse. No horizontal line may cross its graph twice.
  5. y = √x and y = ∛x are the inverses of squaring (x ≥ 0) and cubing, reflected across y = x.
◇ FAMILY RULE CARDS
Write five rule cards, each with two steps, like "times 3, then subtract 2." Keep the numbers friendly.
Deal a card. One person runs a number forwards. Another takes the output and runs it backwards to recover the number.
Write the inverse rule for each card as a function. Check that forwards then backwards gives the number back.
For the trick round, add a card that reads "square it." Decide together whether it has an inverse, and why.
For a grown-up
This is a paper activity. If the family repeats the drip lab, a grown-up sets the tap and uses cold water only.
A wrong answer is a good moment: ask which step should be undone first. Inverses undo the last step first.
The drip rule and every number on the cards are made up for the game, not facts about any real tap.
GARDEN DAY CHECK
  • Read the question.
  • Tap your answer.
If f(x) = 3x - 2, what is the inverse function?
The card table reads V(0) = 10, V(5) = 30, V(10) = 50, V(15) = 70, V(20) = 90. Backwards from 30: what is V⁻¹(30)?
A card asks: solve √(4t + 1) = 11. What is t?
A card reads "times 4, then add 10." Someone runs it backwards from 50. What number do they get? Type the number.
WHY THIS EXERCISEAn inverse undoes the steps in reverse order. That is the whole method for building one.
Draw two of the family cards with their rules, and write the inverse rule on the back of each.
← Friday