Question

what's domain and range

Answer 1

Domain and range refer to the input and output values of a function.

- Domain: the set of all input values (typically x) for which the function is defined.
- Range (or image): the set of all output values (typically y or f(x)) the function actually takes.

Notation: Domain of f often written Dom(f) or D(f); range written Range(f) or Im(f).

Quick rules to find the domain:
- Exclude values that make denominators zero.
- For even roots, require the radicand ≥ 0.
- For logarithms, require the argument > 0.
- Otherwise, include all real numbers if no restriction applies.

Ways to find the range:
- Solve y = f(x) for x in terms of y and determine which y give real/allowed x.
- Use algebraic manipulation (completing the square, rational inversion).
- Use graphing or calculus (find minima/maxima and behavior at infinity).

Examples:
- f(x) = x^2
- Domain: all real numbers R
- Range: [0, ∞)

- f(x) = 1/(x − 2)
- Domain: R \ {2}
- Range: R \ {0} (the function never equals 0)

- f(x) = √(x − 3)
- Domain: [3, ∞)
- Range: [0, ∞)

- f(x) = ln(x)
- Domain: (0, ∞)
- Range: (−∞, ∞)

- f(x) = sin x
- Domain: R
- Range: [−1, 1]

Note: The codomain is the set you declare as possible outputs; the range/image is the subset of the codomain that the function actually attains. If you want, tell me a specific function and I’ll find its domain and range.