Question

If $f\left( x \right)=\dfrac{\left[ x \right]}{\left[ x-1 \right]},x\ne 1$ then ${{\left( fofof.......f \right)}_{\left( 17\,times \right)}}\left( x \right)$ is equal to
A) $\dfrac{\left[ x \right]}{\left[ x-1 \right]}$
B) $x$
C) {{\left\\{ \dfrac{x}{\left[ x-1 \right]} \right\\}}^{17}}
D) $\left[ \dfrac{17x}{\left( x-1 \right)} \right]$

Answer 1

Here we have to find the value of function composition given by using the value of the function given. Firstly we will find the value of $\left( fof \right)\left( x \right)$ by using the function composition rule. Then we will find the value of $\left( fofof \right)\left( x \right)$ which has the function inside the function two times. Finally by these two values we will seek a pattern and get the desired answer.

Complete answer: The function is given as follows:
$f\left( x \right)=\dfrac{\left[ x \right]}{\left[ x-1 \right]},x\ne 1$…..$\left( 1 \right)$
We have to find the value of,
${{\left( fofof.......f \right)}_{\left( 17\,times \right)}}\left( x \right)$
So we will start by taking two functions as follows,
$\left( fof \right)\left( x \right)=f\left( f\left( x \right) \right)$….$\left( 2 \right)$
On putting the value of function above we get,
$\left( fof \right)\left( x \right)=f\left( \dfrac{\left[ x \right]}{\left[ x-1 \right]} \right)$
Now using the value in equation (1) we get,
$\left( fof \right)\left( x \right)=\dfrac{\left[ \dfrac{\left[ x \right]}{\left[ x-1 \right]} \right]}{\left[ \dfrac{\left[ x \right]}{\left[ x-1 \right]}-1 \right]}$
$\Rightarrow \left( fof \right)\left( x \right)=\dfrac{\left[ \dfrac{\left[ x \right]}{\left[ x-1 \right]} \right]}{\left[ \dfrac{\left[ x \right]-\left[ x-1 \right]}{\left[ x-1 \right]} \right]}$
On solving further we get,
$\Rightarrow \left( fof \right)\left( x \right)=\dfrac{\left[ \dfrac{\left[ x \right]}{\left[ x-1 \right]} \right]}{\left[ \dfrac{1}{\left[ x-1 \right]} \right]}$
$\left( fof \right)\left( x \right)=x$
Put the above value in equation (2),
$f\left( f\left( x \right) \right)=x$….$\left( 3 \right)$
Next we will find the value of,
$\left( fofof \right)\left( x \right)=f\left( f\left( f\left( x \right) \right) \right)$
Put the value from equation (3) above we get,
$\Rightarrow \left( fofof \right)\left( x \right)=f\left( x \right)$
$\Rightarrow \left( fofof \right)\left( x \right)=\dfrac{\left[ x \right]}{\left[ x-1 \right]}$
So as we can see that when the function composition of even number of function is found it is $x$ and when the function composition of odd numbers of function are found it is $\dfrac{\left[ x \right]}{\left[ x-1 \right]}$ .
Therefore the function composition found $17$ times will be equal to $\dfrac{\left[ x \right]}{\left[ x-1 \right]}$ as $17$ is an odd number.
Hence the correct option is (A).

Note:
Function composition is an operation that takes two functions and produces a third function using them. It is a special case of composition of relations. It is not like multiplication; it is different as the variable of the function is another function and it is substituted in the first function to get the outcome. It is tricky when we say it like that but once we know how to evaluate such composition we see it is quite easy.