Let N be the set if natural numbers and two functions f and g be difined as f,g :N\rightarrow Nsuch that

f(n) = \left\{\begin{matrix} \frac{n+1}{2}\;if\;n\;is\;odd\\ \frac{n}{2}\;if\;n\;is\;even \end{matrix}\right. and g(n) = n - (-1)^n . Then fog is:

  • Option 1)

    Onto but not one-one

  • Option 2)

    Neither one-one nor onto

  • Option 3)

    one-one but not onto

  • Option 4)

    both one-one and onto

Answers (1)
A admin

 

Onto function -

If  f:A\rightarrowB is such that each & every element in B is the f image of atleast one element in A.Then it is Onto function.

 

- wherein

The range of f is equal to Co - domain of f.

 

 

One - One or Injective function -

A function in which every element of range of function corresponds to exactly one elements.

- wherein

A line parallel to x - axis cut the curve at most one point.

f(n)=\left\{\begin{matrix} \frac{n+1}{2} \: , \: n \: \: is\: \: odd\\ \frac{n}{2} ,\: n\: \: is\: \: even \end{matrix}\right.

g(n)=n-(-1)^{n}=\left\{\begin{matrix} n+1; n\: \: is\: \: odd\\ n-1; n\: \: is \: \: even \end{matrix}\right.

f(g(n))=n-(-1)^{n}=\left\{\begin{matrix} \frac{n-1}{2};\: n\: \: is\: \: odd\\ \frac{n}{2};\: n\: \: is \: \: even \end{matrix}\right.

\therefore Many one but onto function

 

 

 

 


Option 1)

Onto but not one-one

Option 2)

Neither one-one nor onto

Option 3)

one-one but not onto

Option 4)

both one-one and onto

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