Question

Assertion :Let $f$ and $g$ be increasing and decreasing functions respectively from $[0,\infty ]$ to $[0,\infty ]$. Let $h\left(x\right)=f\left(g\right(x\left)\right)$. If $h\left(0\right)=0,$ then $h\left(x\right)$ is always zero Reason: $h\left(x\right)$ is an increasing function of $x$

• 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 C Assertion is correct but Reason is incorrect

Since $g\left(x\right)$ is decreasing,

$\therefore g\left(x_2\right)\le g\left(x_1\right)$ when $x_2\ge x_1$

Since $f\left(x\right)$ is increasing,

$\therefore f\left[g\left(x_1\right)\right]\ge f\left[g\left(x_2\right)\right]$

$\Rightarrow h\left(x_1\right)\ge h\left(x_2\right)$ when $x_2\ge x_1.$

$\Rightarrow h\left(x\right)$ is a decreasing function of $x$ and $h\left(0\right)=0.$

Also, domain of $h=[0,\infty )$ and range of $h=[0,\infty )$.

$\therefore h\left(x\right)=0,\forall x\in [0,\infty ).$