Mathematics –Class XII

_{Unit-I:Relations and functions}

​

Chapter-1: Relations and functions

Sub Topic: Type of function

Outline

• One-one function
• Many one function
• Onto function
• Into function
• Domain and Range
• Assignment

M.P.

Bihar

Goa

Bhopal

Patna

Panji

M.P.

Click here

Bihar

Click here

Goa

Click here

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

M.P.

Bihar

Goa

Bhopal

Patna

Panji

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

f={(M.P.,Bhopal),(Bihar, Patna),(Goa,Panji)}

One-one function

M.P.

Bihar

Goa

Bhopal

Patna

Panji

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

We concluded that f is one-one function ,if distint elements have dintinct image in co-domain

One-one function

Hence, f(x₁ )= f(x₂ )=> x₁ = x₂ OR

x₁ ≠ x₂ =>f(x₁ ) ≠ f(x₂ )

M.P.

Bihar

Bhopal

Patna

Panji

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

f={(M.P.,Bhopal),(Bihar,Patna),(Haryana,Chandigarh),

(Punjab,Chandigarh)}

Many-one function

Punjab

Chandigarh

Haryana

M.P.

Bihar

Bhopal

Patna

Panji

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

We concluded that f is many-one function ,if distinct elements may have same image in co-domain.

Many-one function

Punjab

Chandigarh

Haryana

Hence function is many –one if

If there exist x₁ = x₂ s . t., x₁ ≠ x₂ =>f(x₁ )= f(x₂ )

M.P.

Bihar

Punjab

Bhopal

Patna

Panji

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

f={(M.P.,Bhopal),(Bihar,Patna),(Haryana,Chandigarh),

(Punjab,Chandigarh)}

Into function

Chandigarh

Haryana

M.P.

Bihar

Punjab

Bhopal

Patna

Panji

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

Into function

Chandigarh

Haryana

We concluded that f is into function ,if there is at least one element in co-domain which do not have pre-image in domain

M.P.

Bihar

Punjab

Bhopal

Patna

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

f={(M.P.,Bhopal),(Bihar,Patna),(Haryana,Chandigarh),

(Punjab,Chandigarh)}

onto function

Chandigarh

Haryana

M.P.

Bihar

Punjab

Bhopal

Patna

X

Y

Here, f: X🡪 Y s.t, y is capital of x.

onto function

Chandigarh

Haryana

Here every element of co domain is associated with some element of domain .i.e., no any element of co domain is free.

It means,∀yϵ co domain ,there exists xϵ domain such that f(x)=yi.e., Co domain =Range of f

Onto:- Any function f:X🡪Y is onto if

∀yϵ co domain ,there exists xϵ domain such that f(x)=y i.e., Co domain =Range of f

One-one:-Any function f:X🡪 Y f is one-one if

f(x)=f(y)=>x=y or x≠y=>f(x)≠f(y)

Many-one:-Any function f:X🡪Y is many-one if If there exist x₁ = x₂ in domain s . t., x₁ ≠ x₂ =>f(x₁ )= f(x₂ )

Into:-Any function f:X🡪Y is into ,if there exists at least one element in co domain which do not have pre-image in domain. i.e.,

Q-2.Show that f:R₊ 🡪 [4, ∞[ ,given by f(x)=x² +4

is one-one and onto.

Q-3.Show that f:R₊ 🡪 [-5, ∞[ ,given by

f(x)=9x² +6x-5 is one-one and onto.

Assignment

M.P.

Bhopal

Back

Bihar

Patna

Back

Goa

Panaji

Back