Question

If f(x)= x + 2 and f(x) o f(x) o f(x)... = q where o represents the composition of two function and n denotes the number of times of composition, the value of n is given by which of the following expressions?

A) n=(q-x)/2
B) n=q/2
C) n=q-x
D) n is any odd number

Answer 1

Given  f(x)= x+ 2

fx o fx o fx =     f   (  f (  fx) ))

we start with   inner most function

we know fx = x+2

that mean  if we input x the function adds up a 2 in it  .

so let us  replace inner most fx with  x+2

it looks like  :      f (  f (  x+2)  )

 now this time inner  function is f( x+2)  and it's input is x +2 so we would add a 2 in  x+2

this becomes  :   f   ( x+2+2 )  = f( x+4)

this time we have the function f( x+4)  for which input is  x+4  so the function will add 2 in it  giving us :  

   x+ 4 + 2   = x+6

we can see fx= x+2

fxo fx = x+4=    x+ 2*2

fxo fxofx = x+6  = x+ 3* 2      this is making a series

fx0fx ofx ...... n times =   x + n* 2

which is given equal to q

So x+ 2n =   =   q

         2n =    q- x

               n=   ( q-x) /2

Answer : option (a)

fxofxofx ............................o   .nth time fx

it would be