Codomain - Usually “Standard” Sets
The codomain is typically one of the well-known, “nice” mathematical sets:
- ℝ (real numbers)
- ℤ (integers)
- ℕ (natural numbers)
- ℂ (complex numbers)
- [0,1] (unit interval)
- ℝⁿ (n-dimensional space)
Range - More Specific/Irregular Sets
The range often ends up being more specific or “messier”:
- [0, ∞) instead of ℝ
- {1, 4, 9, 16, 25, …} (perfect squares) instead of ℕ
- (-1, 1) instead of ℝ
- {even integers} instead of ℤ
Examples
| Function |
Codomain |
Range |
| f(x) = x² |
ℝ |
[0, ∞) |
| f(x) = sin(x) |
ℝ |
[-1, 1] |
| f(x) = 1/x |
ℝ |
ℝ {0} |
| f(n) = 2n |
ℤ |
{even integers} |
Why This Happens
- Convenience: It’s easier to say “f: ℝ → ℝ” than to figure out the exact range first
- Context: We often work within familiar number systems
- Composition: Using standard sets makes function composition cleaner
- Generality: Standard codomains work for families of similar functions
So yes - codomain = “the usual suspects” and range = “what actually happens” is a great way to think about it!