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Algebra 1 9-12 / Week 03 / Monday
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Week 03 · Functions

Monday

The crank box
// One input, exactly one output
⏱ about 20 min

Monday: The Crank Box

Comet sets a cardboard box on the bench. It has a slot on each side and a crank on top. "Function machine."

Wren writes 4 on an index card and feeds it into the left slot. Comet turns the crank. A card slides out: 14.

"Rule: multiply by 3, add 2," Comet says. "Try another." Wren feeds in 5. Out comes 17.

"What do you notice?" Wren asks. "Feed in 4 again."

Comet cranks. 14 again. "Same input, same output. Every time."

Nova hovers over the slot, her light tracing the card's path. "Would you like a hint? That is what makes it a function."

"One input, exactly one output," Wren says. "The inputs we allow are the domain. The outputs are the range."

Comet turns the crank on a cardboard function machine while Wren feeds in a card and Nova hovers.

A function gives exactly one output

A function is a rule that takes an input and gives back exactly one output. Feed in the same input twice, get the same output twice.

The crank box's rule is × 3 + 2. Feed in 4 and 14 comes out. Feed in 5 and 17 comes out.

The set of inputs the box accepts is its domain. The set of outputs it gives is its range.

A rule machine labeled × 3 + 2 with 4 going in and 14 coming out.

A solved problem to study

Here is how Wren found the range of the crank box for the inputs 1, 2, 3 and 4. Read each step.

  1. Write the rule as an expression: the output is 3x + 2 when x goes in.
  2. Put each input in: 1 gives 5, 2 gives 8, 3 gives 11, 4 gives 14.
  3. The domain is {1, 2, 3, 4}. The range is {5, 8, 11, 14}.
  4. Check that no input gave two outputs. It is a function.
Input (x)Output (3x + 2)
15
28
311
414
5?

When a rule is not a function

Suppose the box gave 14 for one card marked 4 and 20 for the next card marked 4. The same input gave two outputs.

That box is not a function. You could not predict its output, so you could not use it in a build.

A rule can give the same output for two different inputs and still be a function. Only repeated inputs with different outputs break it.

CRANK THE BOX
  • Read the question.
  • Tap your answer.
A rule machine labeled × 3 + 2, 5 going in, a blank coming outThe crank box rule is × 3 + 2. What comes out when 5 goes in?
The crank box rule is × 3 + 2. What comes out when 10 goes in?
The crank box rule is × 3 + 2. What comes out when 0 goes in?
FUNCTION OR NOT?
  • Read the question.
  • Tap your answer.
Crank box: (4, 14), (5, 17), (4, 14), (6, 20). Is this relation a function?
Broken box: (4, 14), (5, 17), (4, 20). Is this relation a function?
Flat box: (1, 7), (2, 7), (3, 7). Every input gives 7. Is this a function?
StatementTrue or false?
A function gives exactly one output for each input.?
If two different inputs give the same output, the rule is not a function.?
3 × 4 + 2 = 14?
For the crank box with domain {1, 2, 3, 4}, the range is {5, 8, 11, 14}.?
The domain is the set of outputs a function gives.?
WHY THIS EXERCISEEvery rule you build this year will be a function. Knowing the test keeps your models trustworthy.
Try it
Make a rule card for a machine at home, such as "× 2 + 1". Have someone feed you three inputs.
Write each input and output as a pair. Check that no input gave two outputs.
Draw the crank box with the input 4 going in and 14 coming out. Write the rule on the box.

Strong start. Tomorrow the rule gets a name, and f(3) = 11 tells a story.