First slide

Functions (XII)

Question

$If f (x + \frac{1}{2}) + f (x - \frac{1}{2}) = f (x)$ or all $x \in R$ Then the period of $f (x)$ is

Moderate

A

1
B

2
C

3
D

4

Solution

$(x + \frac{1}{2}) + f (x - \frac{1}{2}) = f (x)$ (1)
Replacing x by $(x + \frac{1}{2})$ , we get
$f (x + 1) + f (x) = f (x + \frac{1}{2})$ (2)
or $f (x + 1) + f (x - \frac{1}{2}) = 0$ (from (1) and (2))
$\begin{array}{l} or f (x + \frac{3}{2}) = - f (x) (Replacing x by (x + \frac{1}{2})) \\ or f (x + 3) = - f (x + \frac{3}{2}) = f (x) (Replacing x by (x + \frac{3}{2})) \end{array}$
Therefore, f(x) is periodic with period 3.

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