Question

what is range and domain

Answer 1

Domain and range are basic parts of a function.

- Domain: the set of all possible input values (x-values) for which the function is defined.
- Range: the set of all actual output values (y-values) the function produces from those inputs.

How to think about them
- If f is a rule that takes x and returns y = f(x), the domain is the allowable x’s and the range is the set of resulting y’s.
- On a graph: domain = projection of the graph onto the x-axis; range = projection onto the y-axis.

Common ways to determine domain
- Look for operations that restrict x: division by zero, even roots (square root requires nonnegative radicand), logarithms (argument > 0), etc.
- Solve algebraically: find x for which the formula makes sense.

Common ways to determine range
- Solve y = f(x) for x and see which y values give real x solutions (this finds the image).
- Use the graph: see which y-values appear on the graph.
- Consider behavior at extremes (limits) and any minimum/maximum values.

Examples
- f(x) = x^2: domain = all real numbers, range = y ≥ 0.
- f(x) = 1/x: domain = x ≠ 0, range = y ≠ 0.
- f(x) = √(x − 3): domain = x ≥ 3, range = y ≥ 0.
- f(x) = ln x: domain = x > 0, range = all real numbers.

Note on codomain vs. range
- Codomain is the set you intend outputs to lie in (a part of the function definition). Range (also called image) is the actual set of outputs. Example: if f: R → R is defined by f(x)=x^2, the codomain is R but the range is [0, ∞).

If you want, tell me a specific function and I’ll find its domain and range.