Function f is symmetric to the origin and periodic with period 8. If f(2)=3, what is the value of f(4)+f(6)?

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Answer 1 · 2018-07-26T19:20:15

#-3#

Let #f(x)# be the function symmetric to the origin i.e. #f(x)# is odd hence

#f(-x)=-f(x)#

Since function #f(x)# is periodic with period #8# hence we have

#f(x+8)=f(x)#

setting #x=-4# in above equation we get

#f(-4+8)=f(-4)#

#f(4)=f(-4)#

#f(4)=-f(4)\quad (\because f(-x)=-f(x))#

#2f(4)=0#

#f(4)=0#

Again setting #x=-2# in above function we get

#f(-2+8)=f(-2)#

#f(6)=-f(2) \quad (\because f(-x)=-f(x))#

#f(6)=-3 \quad (\because f(2)=3)#

#\therefore f(4)+f(6)#

#=0-3#

#=-3#