Let `f(x)=[x]`, where [.] is the greatest integer function and `g(x)=sinx` be two real valued functions over R.
Which of the following statements is correct?
1. (fof)(x)=f(x).
2. (gog)(x)=g(x) only when x=0.
3. (go(fog))(x) con take only three values.
Select the correct answer using the code given below:

Options

A. 1 and 2 olny
B. 2 and 3 only
C. 1 and 3 only
D. 1, 2 and 3

Answer

C

Explanation

`(fof)(x)=[[x]]andf(x)=[x]`
suppose x=0.2
`implies(fof)(x)=[[0.2]]=[0]=0`
`f(x)=[0.2]=0`
`impliesfof(x)=f(x)`
Now, (gog)(x)= sin sin x and g(x) =sin x
At x=0
(gog)(x)=sin sin 0= sin 0=0
`implies(gog)(x)=g(x)" at "x=0`
and this is true for
`x=npi" where "n=0,1,2,3,4......`
`:'(fog)(x)=[sinx]`
`(go(fog))(x)=sin[sinx]`
`:'` sinx has value from -1 to 1
If `-1lesinxlt0`
`(go(fog))(x)=sin(-1)-sin(-1)`
If `-1lesinxlt0`.
`:.(go(fog))(x)=sin(0)=sin0and"If "sinx=1`
`:.(go(fog))(x)=sin(1)=sin1`