Functions Question 707
Question: If $ f:R\to R $ and $ g:R\to R $ are given by $ f(x)=| x | $ and $ g(x)=[x] $ for each $ x\in R, $ then [$x\in R:g(f(x))\le f(g(x))$}=
Options:
A) $ Z\cup (-\infty ,0) $
B) $ (-\infty ,0) $
C) $ Z $
D) R
Show Answer
Answer:
Correct Answer: D
Solution:
[d] $ g(f(x))=g(| x |)=[| x |]; $ $ f(g(x))=f([x])=| [x] | $ When $ x\ge 0,[| x |]=[x]=| [x] | $
$ \therefore ,f(g(x))=g(f(x)) $ When $ x<0,[x]\le x<0\Rightarrow | [x] |\ge | x | $
$ \therefore | [x] |\ge | x |\ge [| x |] $
$ \Rightarrow f(g(x))\ge g(f(x)) $ Thus, $ g(f(x))\le f(f(x)) $ for all $ x\in R $
BETA
