## 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 D_{1} and D_{2} respectively then the domain of f(x) ± g(x) or f(x) . g(x) is D_{1 }∩ D_{2}.

While domain of \(\frac { f(x) }{ g(x) } \) is D_{1 }∩ D_{2} – {g(x) = 0}.

Domain of (√f(x)) = D_{1 }∩ {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