Question

Given is a partial graph of an even periodic function $f$ whose period is 8 . If $\left[{}^{\ast }\right]$ denotes greatest integer function then find the value of the expression.
$f(-3)+2|f(-1)|+\left[f\left(\frac{7}{8}\right)\right]+f(0)+\arccos (f(-2))+f(-7)+f(20)$

Answer 1

Answer: 5

$f(-3)=f(3)=2[f(x)$ is an even function, $\therefore f(-x)=f(x)]$
again $f(-1)=f(1)=-3$
$\therefore 2|f(-1)|=2|f(1)|=2|-3|=6$
from the graph, $-3<f\left(\frac{7}{8}\right)<-2$
$\therefore \left[f\left(\frac{7}{8}\right)\right]=-3$
$f(0)=0\text{ (obviously from the graph) }$
${\cos }^{-1}(f(-2))={\cos }^{-1}(f(2))={\cos }^{-1}(1)=0$
$f(-7)=f(-7+8)=f(1)=-3[f(x)\text{ has period }8]$
$f(20)=f(4+16)=f(4)=3[f(nT+x)=f(x)]$
$\text{ sum }=2+6-3+0+0-3+3$
$\therefore \text{ sum }=5$