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Domain, Co-Domain, and Range

Let f: X Y be a function. In general sets X and Y could be any arbitrary non-empty sets. But at this level we would confine ourselves only to real-valued functions. That means it would be invariably assumed that X and Y are the subsets of real numbers.
 
Set “X” is called domain of the function “f”.
 
Set “Y” is called co-domain of the function “f”.
 
Set of images of different elements of set X is called the range of the function “f”. It is obvious that range could be a subset of co-domain as we may have few elements in co-domain which are not the images of any element of the set X (of course these elements of co-domain will not be included in the range). Range is also called domain of variation. Domain of function “f” is normally represented as Domain (f). Range is represented as Range (f). Note that sometimes domain of the function is not explicitly defined. In these cases domain would mean the set of values of “x” for which f(x) assumes real values, e.g., if y = f(x) then Domain (f) = {x: f(x) is a real number}.

Rules for finding the domain of a function

  1. Domain (f(x) + g(x)) = Domain f(x) Domain g(x).
  2. Domain (f(x) g(x)) = Domain f(x) Domain g(x).
  3. Domain 78286.png f(x) Domain g(x) {x : g(x) ≠ 0}.
  4. Domain 78280.png Domain f(x) {x : f(x) ≥ 0}.
  5. Domain (fog) = Domain (g(x), where fog is defined by fog(x) = f{g(x)}.

Trigonometric functions

Function
Domain
Range
f(x) = sin x
R
[–1, 1]
f(x) = cos x
R
[–1, 1]
f(x) = tan x
R78270.png
R
f(x) = cot x
R – {nπ, n Z}
R
f(x) = sec x
R78264.png
(–∞,– 1] ∪ [1, ∞)
f(x) = cosec x
R – {nπ, n Z}
(–∞,– 1] ∪ [1, ∞)

Important result

f(x) = a cos x + b sin x
 
= 78251.png sin(x + tan–1 78245.png)
 
= 78239.png cos(x – tan–1 78233.png
 
Range of f(x) = a cos x + b sin x is
 
78226.png

Inverse trigonometric functions

Function
Domain
Range
f(x) = sin–1x
[–1, 1]
78215.png
f(x) = cos–1 x
[–1, 1]
[0, π]
f(x) = tan–1 x
R
78209.png
f(x) = cot–1 x
R
(0, π)
f(x) = sec–1 x
(–∞, –1] [1,∞)
[0, π] – {π/2}
f(x) = cosec–1 x
(–∞, –1] [1,∞)
[–π/2, π/2] – {0}




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