Question

Assertion :If $f\left(x\right)$ is continuous then |f(|x|)| is also continuous Reason: If $\left|f\left(x\right)\right|\le \left|x\right|$ then |f(x)| is continuous at $x=0$

• 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. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

If $f\left(x\right)=x^2-5x+6$ then $f\left(\left|x\right|\right)={\left|x\right|}^2-5\left|x\right|+6$
Graph of f(x)
Now $\because \left|f\left(x\right)\right|\le \left|x\right|$
$\Rightarrow$ $\left|f\left(0\right)\right|\le 0$ $\Rightarrow$ $f\left(0\right)=0$
$\because$ $\left|f\left(x\right)\right|\le \left|x\right|$
$\therefore$ ${lim}_{x\to 0}\left|f\left(x\right)\right|\le {lim}_{x\to 0}\left|x\right|$
$\Rightarrow$ ${lim}_{x\to 0}\left|f\left(x\right)\right|\le 0$ but ${lim}_{x\to 0}\left|f\left(x\right)\right|\ge 0$
$\therefore$ ${lim}_{x\to 0}\left|f\left(x\right)\right|=0=\left|f\left(0\right)\right|$
$\Rightarrow |f\left(x\right)|$ is continuous at $x=0.$