Question

Which of the following statements are correct?
1. $\left(fof\right)\left(x\right)=f\left(x\right)$.
2. $\left(gog\right)\left(x\right)=g\left(x\right)$ only when $x=0$.
3. $\left(go\right(fog\left)\right)\left(x\right)$ can take only three values.
Select the correct answer using the code given below

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

Answer 1

Answer: (A)

1 and 2 only

$fof\left(x\right)=\left[\right[x\left]\right]=\left[x\right]=f\left(x\right)$

$gog\left(x\right)=\sin \left(\sin x\right))=g(x)$ at $x=0$ as $g\left(0\right)=\sin \left(\sin 0\right))=0$

$go\left(fog\right)\left(x\right)=\sin \left[\sin x\right]=-1$ or $0$ but not $1$ as the highest value of $go\left(fog\right)\left(x\right)$ can only be $\sin 1$ and not $1$.

Hence, only statement 1 and 2 are correct.