Question

If $f:[1,10]\to [1,10]$ is a non-decreasing function and $g:[1,10]\to [1,10]$ is a non-increasing function. Let $h\left(x\right)=f\left(g\right(x\left)\right)$ with $h\left(1\right)=1$, then $h\left(2\right)$

• A. lies in $(1,2)$
• B. is more than $2$
• C. is equal to $1$
• D. is not defined

Answer 1

Answer: (C)

is equal to $1$

$\begin{array}{c}\Rightarrow f:\left[1,10\right]\to \left[1,10\right]\\ \Rightarrow g:\left[1,10\right]\to \left[1,10\right]\\ \Rightarrow h\left(x\right)=f\left(g\left(x\right)\right)\\ \Rightarrow h\left(1\right)=f\left(g\left(1\right)\right)=f\left(1\right)=1\\ \Rightarrow h\left(10\right)=f\left(g\left(10\right)\right)=f\left(1\right)=1\\ \Rightarrow then,h\left(2\right)=fg\left(2\right)=\left[1\right]\\ \Rightarrow 1\in \left[1,10\right]\\ hence,h\left(2\right)=\left(1\right)\end{array}$