First slide

Functions (XII)

Question

$If f : R \to R and g : R \to R are given by f (x) = | x | and g (x) = [x] for each x \in R, then {x \in R : g (f (x)) \leq f (g (x))} =$

Moderate

A

$Z \cup (- \infty, 0)$
B

$(- \infty, 0)$
C

$Z$
D

$R$

Solution

$f (x) = |x| a n d g (x) = [x] \Rightarrow g (f (x)) = g (|x|) = [|x|] a n d f (g (x)) = f ([x]) = |[x]| w h e n x \geq 0, [|x|] = [x] = |[x]|$
$t h e r e f o r e g (f (x)) = f (g (x)) w h e n x < 0, [x] \leq x < 0 (\sin c e [x] \leq x f o r a l l x) |[x]| \geq |x| \geq [|x|]$
$\Rightarrow f (g (x)) \geq g (f (x)) t h e r e f o r e g (f (x)) \leq f (g (x)) f o r a l l x \in R$

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