Answer

Verified

381.6k+ views

**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 \[G\]’.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Who was the Governor general of India at the time of class 11 social science CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference Between Plant Cell and Animal Cell