Question Number 66116 by Rio Michael last updated on 09/Aug/19
$${Given}\:{that}\:\:{f}\left({x}\right)\:=\:\begin{cases}{{x},\:\:{for}\:\mathrm{0}\leqslant{x}<\mathrm{2}}\\{\mathrm{0},\:{for}\:\mathrm{2}\leqslant{x}\leqslant\mathrm{3}}\end{cases} \\ $$$${is}\:{periodic}\:{with}\:{period}\:\mathrm{3}\:{units}, \\ $$$${find}\:{the}\:{value}\:{of}\:\:{f}\left(\mathrm{5}\right)\:{and}\:{f}\left(−\mathrm{5}\right) \\ $$$${sketch}\:{the}\:{graph}\:{of}\:{f}\left({x}\right)\:{for}\:{x}\:{between}\:−\mathrm{3}\:{and}\:\mathrm{6} \\ $$$$ \\ $$$${please}\:{i}\:{really}\:{need}\:{explanations}\:{when}\:{solving}\:{the}\:{first}\:{part}\:{of}\:{the}\:{question} \\ $$$${thanks} \\ $$
Commented by Prithwish sen last updated on 09/Aug/19
$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{3}\right)=\mathrm{f}\left(\mathrm{x}−\mathrm{3}\right)\:\because\mathrm{f}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{with}\:\mathrm{3}\:\mathrm{unit} \\ $$$$\because\:\mathrm{f}\left(\mathrm{5}\right)\:=\:\mathrm{f}\left(\mathrm{8}\right)\:\mathrm{or}\:\mathrm{f}\left(\mathrm{2}\right)=\:\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{f}\left(−\mathrm{5}\right)\:=\:\mathrm{f}\left(−\mathrm{2}\right)\:\mathrm{or}\:\mathrm{f}\left(−\mathrm{8}\right)=\:\mathrm{x}=−\mathrm{5} \\ $$
Commented by Rio Michael last updated on 09/Aug/19
$${thank}\:{you} \\ $$
Commented by Rio Michael last updated on 09/Aug/19
$${but}\:{i}\:{dont}\:{the}\:{fact}\:{that}\: \\ $$$${f}\left({x}+\mathrm{3}\right)={f}\left({x}−\mathrm{3}\right)\:{why}\:? \\ $$$${and}\:{your}\:{solving} \\ $$
Commented by Prithwish sen last updated on 09/Aug/19
$$\because\:\mathrm{it}\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{funtion}\:\mathrm{of}\:\mathrm{3}\:\mathrm{unit}\:\mathrm{it}\:\mathrm{will}\:\mathrm{repeat}\:\mathrm{itself} \\ $$$$\mathrm{after}\:\mathrm{or}\:\mathrm{before}\:\mathrm{every}\:\mathrm{3}\:\mathrm{unit}. \\ $$$$\mathrm{that}\:\mathrm{is}\:\mathrm{why}\:\mathrm{f}\left(\mathrm{x}−\mathrm{3}\right),\mathrm{f}\left(\mathrm{x}\right),\mathrm{f}\left(\mathrm{x}+\mathrm{3}\right)\:\mathrm{will}\:\mathrm{have}\:\mathrm{the} \\ $$$$\mathrm{same}\:\mathrm{value}. \\ $$
Commented by Rio Michael last updated on 09/Aug/19
$${thank}\:{you}\:{sir} \\ $$