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Question Number 66116 by Rio Michael last updated on 09/Aug/19

Given that f(x) = { ((x, for 0≤x<2)),((0, for 2≤x≤3)) :} is periodic with period 3 units, find the value of f(5) and f(−5) sketch the graph of f(x) for x between −3 and 6 please i really need explanations when solving the first part of the question thanks

$${Given}\:{that}\:\:{f}\left({x}\right)\:=\:\begin{cases}{{x},\:\:{for}\:\mathrm{0}\leqslant{x}<\mathrm{2}}\\{\mathrm{0},\:{for}\:\mathrm{2}\leqslant{x}\leqslant\mathrm{3}}\end{cases} \\ $$ $${is}\:{periodic}\:{with}\:{period}\:\mathrm{3}\:{units}, \\ $$ $${find}\:{the}\:{value}\:{of}\:\:{f}\left(\mathrm{5}\right)\:{and}\:{f}\left(−\mathrm{5}\right) \\ $$ $${sketch}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)\:{for}\:{x}\:{between}\:−\mathrm{3}\:{and}\:\mathrm{6} \\ $$ $$ \\ $$ $${please}\:{i}\:{really}\:{need}\:{explanations}\:{when}\:{solving}\:{the}\:{first}\:{part}\:{of}\:{the}\:{question} \\ $$ $${thanks} \\ $$

Commented byPrithwish sen last updated on 09/Aug/19

f(x)=f(x+3)=f(x−3) ∵f is a periodic with 3 unit ∵ f(5) = f(8) or f(2)= 0 and f(−5) = f(−2) or f(−8)= x=−5

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{3}\right)=\mathrm{f}\left(\mathrm{x}−\mathrm{3}\right)\:\because\mathrm{f}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{with}\:\mathrm{3}\:\mathrm{unit} \\ $$ $$\because\:\mathrm{f}\left(\mathrm{5}\right)\:=\:\mathrm{f}\left(\mathrm{8}\right)\:\mathrm{or}\:\mathrm{f}\left(\mathrm{2}\right)=\:\mathrm{0} \\ $$ $$\mathrm{and}\:\mathrm{f}\left(−\mathrm{5}\right)\:=\:\mathrm{f}\left(−\mathrm{2}\right)\:\mathrm{or}\:\mathrm{f}\left(−\mathrm{8}\right)=\:\mathrm{x}=−\mathrm{5} \\ $$

Commented byRio Michael last updated on 09/Aug/19

thank you

$${thank}\:{you} \\ $$

Commented byRio Michael last updated on 09/Aug/19

but i dont the fact that f(x+3)=f(x−3) why ? and your solving

$${but}\:{i}\:{dont}\:{the}\:{fact}\:{that}\: \\ $$ $${f}\left({x}+\mathrm{3}\right)={f}\left({x}−\mathrm{3}\right)\:{why}\:? \\ $$ $${and}\:{your}\:{solving} \\ $$

Commented byPrithwish sen last updated on 09/Aug/19

∵ it is a periodic funtion of 3 unit it will repeat itself after or before every 3 unit. that is why f(x−3),f(x),f(x+3) will have the same value.

$$\because\:\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{funtion}\:\mathrm{of}\:\mathrm{3}\:\mathrm{unit}\:\mathrm{it}\:\mathrm{will}\:\mathrm{repeat}\:\mathrm{itself} \\ $$ $$\mathrm{after}\:\mathrm{or}\:\mathrm{before}\:\mathrm{every}\:\mathrm{3}\:\mathrm{unit}. \\ $$ $$\mathrm{that}\:\mathrm{is}\:\mathrm{why}\:\mathrm{f}\left(\mathrm{x}−\mathrm{3}\right),\mathrm{f}\left(\mathrm{x}\right),\mathrm{f}\left(\mathrm{x}+\mathrm{3}\right)\:\mathrm{will}\:\mathrm{have}\:\mathrm{the} \\ $$ $$\mathrm{same}\:\mathrm{value}. \\ $$

Commented byRio Michael last updated on 09/Aug/19

thank you sir

$${thank}\:{you}\:{sir} \\ $$

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