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Question Number 180485 by moh777 last updated on 12/Nov/22

Determine the value of a and b , that make the function f(x) continuity f(x) = { ((x + 3 ,x ≨ 4)),((2ax + b ,x = 4)),((x^2 −3 , x ≩ 4)) :}

$${Determine}\:{the}\:{value}\:{of}\:{a}\:{and}\:{b}\:, \\ $$$$\:{that}\:{make}\:{the}\:{function}\:{f}\left({x}\right)\:{continuity} \\ $$$${f}\left({x}\right)\:=\:\begin{cases}{{x}\:+\:\mathrm{3}\:\:\:\:\:,{x}\:\lneqq\:\mathrm{4}}\\{\mathrm{2}{ax}\:+\:{b}\:\:\:\:,{x}\:=\:\mathrm{4}}\\{{x}^{\mathrm{2}} −\mathrm{3}\:\:\:,\:{x}\:\gneqq\:\mathrm{4}}\end{cases} \\ $$

Commented by Frix last updated on 12/Nov/22

this is not possible, check the question lim_(x→4^− ) f(x) =7 ≠ 13=lim_(x→4^+ ) f(x) we need a vertical line to connect both but a vertical line is not a function the equation of this would be x=4, 7≤y≤13

$$\mathrm{this}\:\mathrm{is}\:\mathrm{not}\:\mathrm{possible},\:\mathrm{check}\:\mathrm{the}\:\mathrm{question} \\ $$$$\underset{{x}\rightarrow\mathrm{4}^{−} } {\mathrm{lim}}\:{f}\left({x}\right)\:=\mathrm{7}\:\neq\:\mathrm{13}=\underset{{x}\rightarrow\mathrm{4}^{+} } {\mathrm{lim}}\:{f}\left({x}\right) \\ $$$$\mathrm{we}\:\mathrm{need}\:\mathrm{a}\:\mathrm{vertical}\:\mathrm{line}\:\mathrm{to}\:\mathrm{connect}\:\mathrm{both}\:\mathrm{but} \\ $$$$\mathrm{a}\:\mathrm{vertical}\:\mathrm{line}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{function} \\ $$$$\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{this}\:\mathrm{would}\:\mathrm{be} \\ $$$${x}=\mathrm{4},\:\mathrm{7}\leqslant{y}\leqslant\mathrm{13} \\ $$

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