First slide

Functions (XII)

Question

Let $f (x) = \frac{x - [x]}{1 + x - [x]},$ where $[x]$ denotes the greatest integer less than or equal to $x,$ then the range of $f$ is

Moderate

A

$[0, 1 / 2]$
B

$[0, 1)$
C

$[0, 1 / 2)$
D

$[0, 1]$

Solution

If $x$ is an integer, $f (x) = 0$
if $x$ is not an integer $x - [x] = \frac{p}{q}$ where $p$ and $q$ are positive integers and $p < q,$ so that $f (x) = \frac{p}{p + q} < \frac{1}{2}$ so $0 \leq f (x) < \frac{1}{2}$ and the required range is $[0, \frac{1}{2}]$

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