Can a constant output be a function?
Compare proportional, offset, constant and squaring rules using the same criterion.
Begin with the activity ↓Can a constant output be a function?
Make a rule giving output three for every input from zero through four. The output does not change, but each input has exactly one output. This is a function too. Being a function does not require an increasing output.
Make a rule giving output three for every input from zero through four. The output does not change, but each input has exactly one output. This is a function too. Being a function does not require an increasing output.
A function with a fixed output is called a constant function. It can be written f(x)=3 with no x on the right. The rule returns three regardless of the allowed input.
The proportional rule f(x)=2x gives zero at zero, and doubling input doubles output. The offset rule f(x)=2x+1 is also a function but not direct proportion. Direct proportion is one type within the broader family of functions.
Another rule multiplies the input by itself: f(x)=x×x, shortened to f(x)=x². The square x² does not mean twice x. Input three gives 3×3=9. We use only integers zero through four here, so negative multiplication is not required.
All four rules assign one output per input, yet their table values, point patterns and output increments differ. Requiring every function to be a straight line or direct proportion would miss constant and squaring rules.
Switch rules below and compare their outputs at the same input. The graph shows only the five integer-domain points. Next we fix an output and ask whether there is one input, several or none.
Listen one paragraph at a time
Hold on to this distinction
The function condition is not a particular speed of change or shape. Check one output per allowed input. The square x² is x×x, while 2x is 2×x.
Change it and check
Choose constant, proportional, offset or squaring rules and compare their points at the same input. Explain why a fixed output can still be a function.
f(x) = 2x + 1 · Domain: integers 0,1,2,3,4
| x | f(x) |
|---|---|
| 0 | 1 |
| 1 | 3 |
| → 2 | 5 |
| 3 | 7 |
| 4 | 9 |
f(2) = 5 · (2, 5)
All four rules are functions. Function does not mean direct proportion.
All four rules have domain integers 0,1,2,3,4 and codomain integers. This is a first comparison, not a full course on linear and quadratic functions.
Try with paper and objects
Write 0,1,2,3,4 in a top row and three under each. In another table square each input and explain why both tables define functions.
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
16 · output value
4² = 4×4 = 16, distinct from 2×4 = 8.
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
Distinguish evaluating a function from finding inputs meeting an output condition.