Relations and Functions — MATHS STD 11 Science — Question

2. 1

Solution:

Let T is a positive real number.

Let f(x) is periodic with period T.

Now, f(x + T) = f(x), for all $\text{x} \in \text{R}$

⇒ x + T - [x + T] = x - [x], for all $ \text{x} \in \text{R}$

⇒ [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