Question

If $f\left(x\right)$ is an odd periodic function with period $2$, then $f\left(4\right)$ is equal to.

• A. $-4$
• B. $4$
• C. $2$
• D. $0$

Answer 1

The correct option is D

$0$

Explanation for correct option:

Find the value of $f\left(4\right)$:

Given,
$f\left(x\right)$ is an odd periodic function

Period $=2$

For odd function, we know that,

$$\begin{gathered} f(-x)=–f\left(x\right) \\ f\left(0\right)=–f\left(0\right) \\ 2f\left(0\right)=0 \\ f\left(0\right)=0 \end{gathered}$$ and $$\begin{gathered} f\left(x+2\right)=f\left(x\right) \\ \Rightarrow f{\left(2\right)}_{x=0}=f\left(0\right) \\ \Rightarrow f\left(2\right)=0 \end{gathered}$$

Since, $f\left(x+2\right)=f\left(x\right)$for all $x\in R$,

Therefore, $f\left(4\right)=f\left(2\right)=0$
Hence, the correct option is D.