Wren spreads four tables across the bench. "Cart runs, cup towers, Nova's mystery and the launch log."
"Cup tower: c(r) = 3r + 1," Comet reads. "Could r be 2.5?"
"Two and a half rows?" Wren says. "What do you notice? Rows come whole. The domain is whole numbers."
"And the cart," Comet says. "d(t) = 40t, but the plank ends. After 6 seconds there is no more plank."
"So the domain is 0 to 6 seconds," Wren says. "The rule works for 100, but the build does not."
Nova slides the mystery table forward. "Would you like a hint? Look at the input 1."
"It appears twice with different outputs," Comet says. "Not a function. Nova, you did that on purpose."
A rule might accept any number. The build it describes usually does not.
The cup tower c(r) = 3r + 1 makes sense only for whole numbers of rows: 0, 1, 2 and so on.
The cart d(t) = 40t makes sense only on the plank, from t = 0 to t = 6 seconds.
So the cart's range runs from d(0) = 0 to d(6) = 240 centimeters, and no further.
| Function | Rule | Sensible domain | Why |
|---|---|---|---|
| Cup tower | c(r) = 3r + 1 | whole numbers 0, 1, 2, ... | rows come whole |
| Cart run | d(t) = 40t | 0 ≤ t ≤ 6 | the plank ends after 6 seconds |
| Jug fill | v(t) = 3t + 15 | t ≥ 0, until the jug is full | time runs forward and the jug has a top |
Here are the four tables from the bench. Every number is the crew's own, made up for the story.
| Table | Pairs (input, output) | Function? |
|---|---|---|
| Cart run (seconds, centimeters) | (0, 0), (1, 40), (2, 80), (3, 120) | yes |
| Cup tower (rows, cups) | (1, 4), (2, 7), (3, 10), (4, 13) | yes |
| Nova's mystery table | (1, 5), (2, 7), (1, 9), (3, 11) | ? |
| Launch log (pumps, height in centimeters) | (4, 120), (6, 180), (4, 150), (8, 240) | ? |
To test a table, look down the input column. If any input appears twice with different outputs, it is not a function.
A mapping diagram shows the same thing with arrows from inputs to outputs. Two arrows from one input means not a function.
| Statement | True or false? |
|---|---|
| Cart run (seconds, centimeters), (0, 0), (1, 40), (2, 80), (3, 120), is a function. | ? |
| Nova's mystery table, (1, 5), (2, 7), (1, 9), (3, 11), is a function. | ? |
| For d(t) = 40t, t = 100 is in the sensible domain of the cart run. | ? |
| The domain of the cup tower is whole numbers because rows come whole. | ? |
| 40 × 6 = 240 | ? |
Sharp reading. Tomorrow functions show up everywhere, and you review the week with graphs.