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Question: If \(f:R \rightarrow R\) and \(f^{- 1}(x) = \frac{x + 5}{3}.\) are given by \(f(x) = |x|\)and \(g(x)...

If f:R→Rf:R \rightarrow R and f−1(x)=x+53.f^{- 1}(x) = \frac{x + 5}{3}. are given by f(x)=∣x∣f(x) = |x|and g(x)=[x]g(x) = \lbrack x\rbrack for each x∈R,x \in R, then {x∈R:g(f(x))≤f(g(x))}=\{ x \in R:g(f(x)) \leq f(g(x))\} =

A

Z∪(−∞,0)Z \cup ( - \infty,0)

B

(−∞,0)( - \infty,0)

C

Z

D

R

Answer

R

Explanation

Solution

g(f(x))≤f(g(x))g(f(x)) \leq f(g(x)) ⇒ g(∣x∣)≤f[x]g(|x|) \leq f\lbrack x\rbrack ⇒ [∣x∣]≤∣[x]∣\lbrack|x|\rbrack \leq |\lbrack x\rbrack|.

This is true for x∈R.x \in R.