Domain, CoDomain, and Range
Let f: X → Y be a function. In general sets X and Y could be any arbitrary nonempty sets. But at this level we would confine ourselves only to realvalued 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 codomain 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 codomain as we may have few elements in codomain which are not the images of any element of the set X (of course these elements of codomain 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
 Domain (f(x) + g(x)) = Domain f(x) ∩ Domain g(x).
 Domain (f(x) ⋅ g(x)) = Domain f(x) ∩ Domain g(x).
 Domain f(x) ∩ Domain g(x) ∩ {x : g(x) ≠ 0}.
 Domain Domain f(x) ∩ {x : f(x) ≥ 0}.
 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

R –

R

f(x) = cot x

R – {nπ, n ∈ Z}

R

f(x) = sec x

R –

(–∞,– 1] ∪ [1, ∞)

f(x) = cosec x

R – {nπ, n ∈ Z}

(–∞,– 1] ∪ [1, ∞)

Important result
f(x) = a cos x + b sin x
= sin(x + tan^{–1} )
= cos(x – tan^{–1}
Range of f(x) = a cos x + b sin x is
Inverse trigonometric functions
Function

Domain

Range

f(x) = sin^{–1}x

[–1, 1]


f(x) = cos^{–1 }x

[–1, 1]

[0, π]

f(x) = tan^{–1 }x

R


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}
