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.