# Domain, Co – Domain and Range

Let f be a relation from A into B then {x ϵ A: (x, y) ϵ R} is codomain of f and {y ϵ B: (x, y) ϵ f} is called range of f

Domain: The domain is input values

Range: The resultant value is called range

Ex: if f = {(a, b), (c, d), (e, f)} then domain f = {a, c, e} and range f = {b, d, f}

Co-domain: The “co-domain” of a function or relation is a set of values that includes the Range as described above, but may also include additional values beyond those in the range

Ex: Ex: Above image having the domain and range the domain is input value and output is range

Example: A simple function like f(x) = x2 can have the domain (what goes in) of just the counting numbers {1, 2, 3, …}, and the range will then be the set {1, 4, 9, …}

Input is {1, 2, 3, …} = domain

Output is {1, 4, 9, …} = range And another function g(x) = x2 can have the domain of integers {…, -3, -2, -1, 0, 1, 2, 3, …}, in which case the range is the set {0, 1, 4, 9, …} Domain and range of trigonometric function

 Trigonometric functions Domain Range sinx (-∞, ∞) [1, -1] cosx (-∞, ∞) [1, -1] tanx R – {(2n + 1)pi/2 (-∞, ∞) cotx R – npi (-∞, ∞) cosecx R – npi (-∞,-1]  È [1,∞) sec R – {(2n + 1)pi/2 (-∞, -1] È [1, ∞)

Ex: Find the domain and range of the real function f defined by f(x) = 4 – x/ x – 4

Sol: Given that f(x) = 4 – x/ x – 4

f (x) = – (x – 4)/ (x – 4)  take a common”– “

f (x) = -1

The range of f(x) is {-1}

Domain is x – 4 ≠ 0

x ≠ 4

The domain is x ϵ R – {4}

Inverse Function: Let f be defined a function from A to B such that for every element of B their exist a image f, f: A → B is a function then {(y, x)} ϵ B x A : (x, y) ϵ f } is called inverse function.it is denoted by f-1.

Ex: if A = {1, 2, 3} and B = {a, b, c} then f : {(1, c), (2, b), (3, a)} is bijection from A into B and f-1  = {(a, 3), (b, 2), (3, a)} is bijection from B into A.

Here we have the function f(x) = 2x + 3, written as The Inverse Function goes the other way So, the inverse of f(x) = 2x + 3 is written
f-1 (y) = $$\frac{\left( y\,-\,3 \right)}{2}$$
Ex: The cool thing about the inverse is that it should give us back the original value: When the function f turns the apple into a banana. Then the inverse function f-1 turns the banana back to the apple