Question

what info do you need to find the domain of a function

Answer 1

Short answer: you need the function’s rule/expression and the set of numbers you’re allowed to use (e.g. real numbers vs. complex numbers). With that you look for operations that forbid certain inputs.

More detail and a simple checklist
- The function formula or definition (explicit expression, piecewise rule, or graph).
- The underlying number system/context (usually real numbers; sometimes integers, complex numbers, etc.).
- Any parameters/constants and whether they have restrictions.

Then examine the formula for restrictions:
1. Division by zero: exclude x values that make any denominator 0.
2. Even roots (square roots, 4th roots, etc.): require the radicand ≥ 0 when working over the reals.
3. Logarithms: require the argument > 0 (for real logs).
4. Fractional exponents with even denominators: treat like even roots (no negative radicand in reals).
5. Inverse trig inputs: e.g. arcsin/arccos require their argument ∈ [−1,1].
6. Functions defined only on integers (factorial, sequences): restrict to integer domain as specified.
7. Piecewise or implicitly defined functions: include x where at least one piece gives a value (or where the implicit definition has a real solution), minus any excluded points.
8. Compositions: x must be in the inner function’s domain and the inner output must lie in the outer function’s domain.
9. Parameter-dependent expressions: consider parameter values that change the domain (e.g. denominator that can become identically 0).

Common pitfalls
- Canceling factors algebraically can hide removable discontinuities — the domain is determined by the original definition (if f(x) = (x−2)/(x−2) given as that formula, x = 2 is excluded even though simplified form is 1).
- Be explicit whether you want the real-domain or complex-domain.

Quick examples
- f(x)=1/(x−2) → domain: all real x ≠ 2.
- g(x)=√(x−3) → domain: x ≥ 3 (over the reals).
- h(x)=ln(x^2−4) → x^2−4 > 0 → x < −2 or x > 2.
- k(x)=√(ln x) → need ln x ≥ 0 → x ≥ 1.

If you want, give me a specific function and tell me whether you mean real or complex domain, and I’ll find its domain.