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Question Number 67684 by Rio Michael last updated on 30/Aug/19

$$giventhefunction$$$$f\left(x\right)=\{\begin{array}{l}{x}^2,for0⩽x<2\\ ax+3,for2⩽x<4\end{array}$$$$isperiodicofperiod4,andiscontinuous.$$$$a)Findthevalueofa.$$$$b)Findthevaluoff\left(6\right)$$$$c)sketchthegraphfory=f\left(x\right).$$$$helpmeplease,forthegraphi{don}^{\prime }tknowwberetoputy=x^{2}andy=ax+3and$$$$wheredoiputacloseddotandanopendot.$$$$$$

Commented by MJS last updated on 30/Aug/19

$$\mathrm{if}f\left(x\right)\mathrm{is}\mathrm{continuous}\mathrm{within}[0;4]\Rightarrow$$$$\Rightarrow x^2=ax+3\mathrm{for}x=2\Rightarrow a=\frac{1}{2}$$$$\mathrm{but}\mathrm{then}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{not}\mathrm{continuous}\mathrm{at}4({\mathrm{it}}^{\prime }\mathrm{s}\mathrm{periodic}\Rightarrow$$$$\Rightarrow \mathrm{it}‘‘\mathrm{starts}{\mathrm{again}}^{″}\mathrm{at}x=4\mathrm{with}\mathrm{the}\mathrm{same}$$$$\mathrm{value}\mathrm{as}\mathrm{at}x=0$$$$f\left(0\right)=0$$$$\underset{x\to 4^{-}}{\lim }f\left(x\right)=\frac{1}{2}\times 4+3=5$$$$\underset{x\to 4^{+}}{\lim }f\left(x\right)=0^2=0$$$$\mathrm{so}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{not}\mathrm{continuous}...$$$$\mathrm{anyway}\mathrm{the}\mathrm{value}\mathrm{of}f\left(6\right)=f(6-4)=f\left(2\right)=4$$$$\mathrm{closed}\mathrm{dots}\mathrm{at}⩽\mathrm{and}⩾$$$$\mathrm{open}\mathrm{dots}\mathrm{at}<\mathrm{and}>$$

Commented by Rio Michael last updated on 14/Sep/19

$$thankyousir,somuch$$

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