Question

Given is a partial graph of an even periodic function $f$ whose period is 8 . If $\left[{ }^*\right]$ 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

$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 $\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 $