Question

Let $f(x+\frac{1}{2})=f\left(x\right)$. If f(2)= 5 and $f\left(\frac{9}{4}\right)=2$. Find f(-3) and $f\left(\frac{1}{4}\right)$.

Answer 1

given $f\left(x\right)=f(x+1/2)$ therefore f(x) is periodic with period 1/2

given $f\left(2\right)=5,f(2+1/4)=2$

now $f(-3)$ can be obtained by subtracting 10 times 1/2 which implies

$f(-3)=f\left(2\right)=5$[since f is periodic with period 1/2]

similarly

$f\left(1/4\right)$ can be obtained by subtracting four times 1/2 from 9/4

which also implies that $f\left(1/4\right)=f\left(9/4\right)=2$ [since $f$ is periodic with period $1/2$]