Domain, co-domain and range of function
If a function f is defined from a set A to set B then for f : A ⟶ B set A is called the domain of function f and set B is called the co-domain of function f. The set of all f-images of the elements of A is called the range of function f.
In other words, we can say
Domain = All possible values of x for which f(x) exists.
Range = For all values of x, all possible values of f(x).
Methods for finding domain and range of function
(i) Domain
Expression under even root (i.e., square root, fourth root etc.) ≥ 0. Denominator ≠ 0.
If domain of y = f(x) and y = g(x) are D1 and D2 respectively then the domain of f(x) ± g(x) or f(x) . g(x) is D1 ∩ D2.
While domain of \(\frac { f(x) }{ g(x) } \) is D1 ∩ D2 – {g(x) = 0}.
Domain of (√f(x)) = D1 ∩ {x : f(x) ≥ 0}
(ii) Range:
Range of y = f(x) is collection of all outputs f(x) corresponding to each real number in the domain.
- If domain ∈ finite number of points ⇒ range ∈ set of corresponding f(x) values.
- If domain ∈ R or R – [some finite points]. Then express x in terms of y. From this find y for x to be defined (i.e., find the values of y for which x exists).
- If domain ∈ a finite interval, find the least and greatest value for range using monotonicity.
Domain and Range of Some Standard Functions
