Question

Show that the function f : NN
defined by f (m) = m² + 2m +3 is
- one​

Answer 1

Step-by-step explanation:

We have, f(x)=x2+x+1

Calculate f(x1):

⇒  f(x1)=x12+x1+1

Calculate f(x1):

⇒  f(x2)=x22+x1+1

Now, f(x1)=f(x2)

⇒  x12+x1+1=x22+x2+1

⇒  x12−x22+x1−x2=0

⇒  (x1−x2)(x1+x2)+x1−x2=0.........  [ Since,

 (a2−b2=(a+b)(a−b) ]

⇒  (x1−x2)(x1+x2+1)=0    

Since, x1+x2+1=0 for any x∈N

∴  x1=x2

So, f is one-one function.

Clearly, f(x)=x2+x+1≥3 for all x∈N

So, f(x) does not assume values 1 and 2.

∴  f is not an onto function.