Question

Assertion :Let $f:R\to R$ be any function. Define $g:R\to R$ by $g\left(x\right)=\left|f\left(x\right)\right|$ for all $x$. Then, $g$ is continuous is $f$ is continuous. Reason: Composition of two continuous functions is a continuous function

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Let $h\left(x\right)=|x|$ for all $x.$

Clearly, $h\left(x\right)$ is continuous for all $x.$

Then, $g\left(x\right)=\left|f\left(x\right)\right|=h\left[f\left(x\right)\right]=\left(hof\right)\left(x\right)$ for all $x.$

Since composition of two continuous functions is continuous, therefore, $g$ is continuous if $f$ is continuous.