Question

f:N to N defined by f(m)=m^2+m+3 is one one function​

Answer 1

Answer------->

Yes , given function is one one

Step-by-step explanation:

Given------> f : N -------> N and

f ( m ) = m² + m + 3

To show------> f ( m ) is one one function

Solution------>

One One function ------> If every element of domain have its unique image in other words no two different elements of domain have same image in codomain then such type of fynction is called one one function .

Now returning to original problem ,

ATQ, f : N ------> N , it means domain and codomain both are set of natural numbers .

f ( m ) = ( m² + m + 3 )

Let , m₁ , m₂ € N it means m₁ and m₂ are in domain ,

f ( m₁ ) = m₁² + m₁ + 3

f ( m₂ ) = m₂² + m₂ + 3

Now , we suppose that ,

f ( m₁ ) = f ( m₂ )

=> m₁² + 2 m₁ + 3 = m₂² + m₂ + 3

=> m₁² + m₁ = m₂² + m₂

=> m₁² - m₂² + m₁ - m₂ = 0

=> ( m₁ + m₂ ) ( m₁ - m₂ ) + 1 ( m₁ - m₂ ) = 0

=> ( m₁ - m₂ ) ( m₁ + m₂ + 1 ) = 0

m₁ + m₂ + 1 ≠ 0 , because m₁ and m₂ are natural numbers and natural numbers always positive

So , it means , m₁ = m₂

It means if images of two elements are same then elements are also same so given function is one one .