Which inputs are allowed?
Distinguish allowed inputs, the output set and the outputs actually reached.
Begin with the activity ↓Which inputs are allowed?
Suppose the table contains only 0→1, 1→2 and 2→2. If card 3 is allowed, its assignment is missing. If only 0, 1 and 2 were allowed, no input is missing. Decide the allowed inputs before judging the mapping.
Suppose the table contains only 0→1, 1→2 and 2→2. If card 3 is allowed, its assignment is missing. If only 0, 1 and 2 were allowed, no input is missing. Decide the allowed inputs before judging the mapping.
The set of allowed inputs is the domain. Here it contains the three numbers 0, 1 and 2. Do not assume that every number is allowed just because you see an expression or diagram. Read the specified domain too.
If the specified output set is 1, 2 and 3, that set is the codomain. Only 1 and 2 are actually reached: these form the range. An unused codomain value such as 3 does not stop the mapping from being a function.
Expanding the domain to 0, 1, 2 and 3 while keeping the table unchanged leaves input 3 missing. Adding 3→3 restores exactly one output per input. Removing an input and completing an assignment are different repairs.
A formula still needs a domain. Notebook counts in our activity are whole numbers, while lengths may allow decimals. The same formula with different allowed inputs defines different functions. The codomain is also part of the specification.
Include or exclude input 3 below and complete its assignment. Distinguish an unused value on the output side from a missing assignment on the input side. Next we compress a verbal rule into an expression.
Listen one paragraph at a time
Hold on to this distinction
Not every codomain value needs an incoming arrow. Each domain input must have exactly one outgoing assignment ending in the codomain.
Change it and check
Change whether input 3 is allowed and whether it is assigned. Read domain, codomain and range, and explain which kind of gap matters.
Domain: 0, 1, 2 · Codomain: 1, 2, 3
| Input | Output |
|---|---|
| 0 | 1 |
| → 1 | 2 |
| 2 | 2 |
1 → 2
Function: each allowed input has exactly one output.
Reached outputs: 1, 2 · Range
This diagram specifies entire finite sets and their assignments. An excluded input has no value under this function.
Try with paper and objects
Draw inputs 0, 1, 2 and output candidates 1, 2, 3 with assignments 0→1, 1→2, 2→2. Leave output 3 unused, then draw what is needed if input 3 is added.
Check and explain another example
Check your idea 1
Check your idea 2
Apply it in a new setting
Write an answer yourself
Return to the explanationCompare with a solution
2 · distinct values
Only 1 and 2 are reached, giving two distinct values.
Does your explanation include these ideas?
Explain aloud or in a drawing, then check your ideas. Different wording can express the same meaning.
Continue, or pause here
Explain variables and function-value notation, then substitute consistently.