Relations and Functions — Maths STD 11 Science — Question
CBSE BoardEnglish MediumSTD 11 ScienceMathsRelations and Functions1 Mark
MCQ
The function $f(x) = x - [x]$ has period of:
✓
$1$
B
$2$
C
$3$
D
$4$
✓
Answer
Correct option: A.
$1$
Let $T$ is a positive real number.
Let $f(x)$ is periodic with period $T.$
Now, $f(x + T) = f(x),$ for all $\text{x} \in \text{R}$
$\Rightarrow x + T - [x + T] = x - [x],$ for all $ \text{x} \in \text{R}$
$\Rightarrow [x + T] - [x] = T,$ for all $ \text{x} \in \text{R}$
Thus, there exist $T > 0$ such that $f(x + T) = f(x)$ for all $\text{x} \in \text{R}$
Now, the smallest value of $T$ satisfying $f(x + T) = f(x)$ for all $\text{x} \in \text{R}$ is $1$
So, $f(x) = x - [x]$ has period $1$
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