# Let N be the set if natural numbers and two functions $f$ and $g$ be difined as $f,g :N\rightarrow N$such 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-oneOption 2)Neither one-one nor ontoOption 3)one-one but not ontoOption 4)both one-one and onto

Onto function -

If  f:A$\rightarrow$B 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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