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  4. Find the period of the function satisfying the relation f(x)+f(x + 3)=0∀ x∈ R. <gwmw style=display:none;></gwmw>

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Q. Find the period of the function satisfying the relation $f\left(x\right)+f\left(x + 3\right)=0\forall x\in R.$

NTA AbhyasNTA Abhyas 2022

Solution:

Given $f\left(x\right)+f\left(x + 3\right)=0$
Replace $x$ by $x+3$
Therefore, $f\left(x + 3\right)+f\left(x + 6\right)=0$
From $\left(1\right)$ and $\left(2\right)$ ,
$f\left(x\right)=f\left(x + 6\right)$
Hence, the function has period $6$