Sets, Relations & Functions
Functions
nta_abhyas_2025
Grade None
Question:
If $f(x)$ is a real valued function such that $f(x+6) - f(x+3) + f(x) = 0, \forall x \in \mathbb{R}$, then period of $f(x)$ is
Step-by-Step Solution
Key Concept: Use functional equation substitutions systematically to derive a period relation $f(x + 18) = f(x)$.
Starting with the functional equation and substituting $x = x + 3$, we obtain $f(x + 9) - f(x + 6) + f(x + 3) = 0$. Adding this to the original equation gives $f(x + 9) + f(x) = 0$. Substituting $x = x + 9$ into this relation yields $f(x + 18) = f(x)$, which establishes that $f$ is periodic with period $18$.
Correct Answer: 3