Question

Assertion :Function $f\left(x\right)=\sin (x+3\sin x)$ is periodic. Reason: If $g\left(x\right)$ is periodic, then $f\left(g\right(x\left)\right)$ may or may not be periodic.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

$f\left(x\right)$ is a periodic function (graph is given below).

Now consider a periodic function $g\left(x\right)$ which has a period of T.
Then

$g\left(x\right)=g(T+x)$.

Now let

$h\left(x\right)=f\left(g\right(x\left)\right)$.

Then

$h(x+T)=f\left(g\right(x+T\left)\right)$ however

$g\left(x\right)=g(x+T)$

Then

$h(x+T)=f\left(g\right(x+T\left)\right)=f\left(g\right(x\left)\right)=h\left(x\right)$

Or
$h(x+T)=h\left(x\right)$.

Hence $h\left(x\right)$ is also a periodic function.

Hence reason is incorrect.