D(f)Domain and Range Calculator
The domain of a function is the complete set of input values (x) for which the function is defined and produces a real output. The range is the complete set of output values (y). Identifying domain and range restrictions is a fundamental skill in algebra and calculus — it determines where functions are valid and where operations such as integration or root-finding apply.
- 1√(expression): domain requires expression ≥ 0
- 2ln(expression): domain requires expression > 0 (strictly positive)
- 31/expression: domain excludes values where expression = 0
- 4Linear ax+b: domain is all real numbers (−∞, +∞)
- 5Composite functions: apply restrictions at each step
f(x) = √(x−3)=Domain: x ≥ 3; Range: y ≥ 0x−3 must be non-negative
f(x) = ln(2x+4)=Domain: x > −2; Range: all reals2x+4 > 0 → x > −2
f(x) = 1/(x²−4)=Domain: x ≠ ±2; Range: y ≠ 0Denominator cannot be zero
| Function Type | Restriction | Domain |
|---|---|---|
| √(ax+b) | ax+b ≥ 0 | x ≥ −b/a |
| ln(ax+b) | ax+b > 0 | x > −b/a |
| 1/(ax+b) | ax+b ≠ 0 | x ≠ −b/a |
| ax+b (linear) | None | All reals (−∞, ∞) |
| xⁿ (even n) | None | All reals |
| arcsin(x) | −1 ≤ x ≤ 1 | [−1, 1] |
| tan(x) | x ≠ π/2 + nπ | All reals except odd multiples of π/2 |
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