Definition: If each element of a set A is associated with exactly one element of a set B, then this association is called a function from A to B.
The set A is called the Domain and the set B is called the Co-domain of the function.
Domain = {1, 2, 3, 4, 5}
Co-domain = {a, b, c, d, e, f}
Range = {a, b, c, d, f}
⇒ Total no. of functions from A to B is
n × n × ..... m times = nm
⇒ Each element of A must be associated.
⇒ All the elements of the set B need not have the association.
⇒ The set of elements of B which are associated with elements of set A is called the Range.
⇒ Range is a subset of the Co-domain
Different types of functions
One-one function: A function f : A → B is called a one-one (or) Injective (or) Monomorphic function if different elements of A have different images in B.
Note: If f is not one-one then it is called a Many-one function.
Onto-function: A function f : A → B is called an onto-function if every element of B is an image of some element of A i.e. if Co-domain = Range
Note: If f is not surjective (onto) then it is called an into function.
Bijective function: A function f : A → B is said to be bijective (or) one-one onto function from A onto B. If f : A → B is both one-one function and onto function.
No. of bijective mappings: If n(A) = m = n(B), then the number of possible bijections from A to B is m!
Constant function: A function f : A → B is called a constant function, if the range of f consists of only one element i.e. , f(x) = k where k is the fixed element of B.
Note: 1. A constant function is a surjection if its Co domain is a singleton set.
2. A constant function is an injection if its domain is a singleton set.
Identity function: The mapping f : A → A defined by f(x) = x, i.e. every element of A is f-image of itself is called Identity function on A. It is denoted by IA (or) I.
Inverse function: If f : A → B is a one-one onto mapping then f-1-1 : B → A is called inverse of the mapping f from A to B.
Note:
Composite function: If f : A → B and g : B → C are two functions then the function gof : A → C defined by (gof)(a) = g[f(a)] , is called the composite (or) product function of f and g.
Odd function: A function f(x) is said to be an add function if f(-x) = - f(x) ∀ x.
Even function: A function f(x) is said to be an even function if f(-x) = f(x)∀ x
Exponential function: If a > 0, a ≠ 1, a ∈ R then the function f(x) = ax is called an exponential function.
Logarithmic function: If a > 0, a ≠ 1, a ∈ R then the function f(x) = logax is called a logarithmic function.
Signum function: It is denoted by y = Sgn(x). It is defined by
The domains and ranges of some standard functions
CONCEPTUAL THEOREMS
Theorem 1: If f : A → B and g : B → C are two one-one (injective) functions then show that the mapping gof : A → C is one-one.
Proof: Given f : A → B and g : B → C are two one-one functions. We have to prove that
gof : A → C is also one-one.
f(a1) = f(a2) [... g is one-one]
a1 = a2 [... f is one-one]
Hence gof: A → C is a one-one function.
Theorem 2: If f : A → B and g : B → C are two onto (Surjective) functions then show that the mapping gof : A→ C is onto.
Proof: Given f : A → B and g : B → C are two onto functions.
We have to prove that gof : A → C is also onto.
Let a ∈ A, b ∈ B, c ∈ C
Since g : B → C is an onto function.
Therefore, there exists an element b ∈ B such that g(b) = c.
Since f : A → B is an onto function.
Therefore, there exists an element a ∈ A such that f(a) = b
Now g(b) = c
g[f(a)] = c
(gof)(a) = c
Thus for any element c ∈ C, there is an element a ∈ A such that (gof)(a) = c
Hence, gof : A → C is an onto.