Question

Period of sin root x+cos root x

Answer 1

Periodic Function

Definition of a periodic function: A function $f(x)$ is said to be periodic if there exists a positive real number $T$ such that

• $f(x+T)=f(x)$

Given: the function $f(x)=sin(\sqrt{x})+cos(\sqrt{x})$

To find: period of $f(x)$

Solution:

• Given, $f(x)=sin(\sqrt{x})+cos(\sqrt{x})$

• Let $T$ be in positive real number.

• Now $f(x+T)$

• $=sin(\sqrt{x+T})+cos(\sqrt{x+T})$

• This cannot be equal to $f(x)$ in general for any $T\in\mathbb{R}^{+}$.

• We have $f(x)\neq f(x+T)$

• Thus $f(x)$ is non-periodic.

Answer: $sin(\sqrt{x})+cos(\sqrt{x})$ is non-periodic.