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**Hint:**Here we are given a function \[G(n)\] and its relation with another function \[F(n)\] is also given. We have to find the function of that function. We do this by simply getting the value of \[G(n)\] using the relation given above. Then we find the value of \[\left( {GoG} \right)(n)\] by using the obtained value of \[G(n)\]. Then we will solve further and make use of the fact that \[k\] here is odd. In this way we can reach our result.

**Complete step-by-step solution:**

Here we are given the function \[F(n)\] as,

\[F(n) = {( - 1)^{k - 1}} \cdot (n - 1)\]

And we are also given another function \[G(n)\] as, \[G(n) = n - F(n)\]. We will put the value of \[F(n)\] in the previous equation to get the value of \[G(n)\] as,

\[G(n) = n - \left( {{{( - 1)}^{k - 1}} \cdot (n - 1)} \right)\]

We know that \[\left( {GoG} \right)(n)\] can be written as,

\[\left( {GoG} \right)(n) = G\left( {G\left( n \right)} \right)\], we will now put the value of \[G(n)\] obtained from above step as,

\[ \Rightarrow \left( {GoG} \right)(n) = G\left( {n - \left( {{{( - 1)}^{k - 1}} \cdot (n - 1)} \right)} \right)\]

We further solve it as,

\[ \Rightarrow \left( {GoG} \right)(n) = n - {( - 1)^{k - 1}}(n - 1) - {( - 1)^{k - 1}}((n - 1) - {( - 1)^{k - 1}}(n - 1))\]

We know that \[{( - 1)^{k - 1}} = 1\] as \[k\] is odd and \[1\] minus odd is even. So, we use this in above equation and move further ahead as,

\[

\Rightarrow \left( {GoG} \right)(n) = n - (n - 1) - \left( {\left( {n - 1} \right) - \left( {n - 1} \right)} \right) \\

\Rightarrow \left( {GoG} \right)(n) = n - (n - 1) \\

\Rightarrow \left( {GoG} \right)(n) = 1 \]

Hence the value of \[\left( {GoG} \right)(n)\] comes out to be \[1\]

**Hence the correct option is \[1)\].**

**Note:**This is to note that this is not the real value of the function \[\left( {GoG} \right)(n)\]. This value is true only when the value of \[k\] is odd as is the case here. For the even value of \[k\] we might have got another result. Here the function we have found, \[\left( {GoG} \right)(n)\] is called as the function of a function and is read as ‘\[G\] of \[G\]’.

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