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f-x-1-log-2x-5-x-1-2-Find-the-domain-of-this-function-




Question Number 204853 by depressiveshrek last updated on 29/Feb/24
f(x)=(√(1−log_((2x+5)) ((x+1)^2 )))  Find the domain of this function
$${f}\left({x}\right)=\sqrt{\mathrm{1}−\mathrm{log}_{\left(\mathrm{2}{x}+\mathrm{5}\right)} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{domain}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function} \\ $$
Answered by Frix last updated on 29/Feb/24
−(5/2)<x≤2∧x≠−2∧x≠−1    We could include the limits  lim_(x→−(5/2))  f(x) =1  lim_(x→−2)  f(x) =(√2)  ⇒ −(5/2)≤x≤2∧x≠−1
$$−\frac{\mathrm{5}}{\mathrm{2}}<{x}\leqslant\mathrm{2}\wedge{x}\neq−\mathrm{2}\wedge{x}\neq−\mathrm{1} \\ $$$$ \\ $$$$\mathrm{We}\:\mathrm{could}\:\mathrm{include}\:\mathrm{the}\:\mathrm{limits} \\ $$$$\underset{{x}\rightarrow−\frac{\mathrm{5}}{\mathrm{2}}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\mathrm{1} \\ $$$$\underset{{x}\rightarrow−\mathrm{2}} {\mathrm{lim}}\:{f}\left({x}\right)\:=\sqrt{\mathrm{2}} \\ $$$$\Rightarrow\:−\frac{\mathrm{5}}{\mathrm{2}}\leqslant{x}\leqslant\mathrm{2}\wedge{x}\neq−\mathrm{1} \\ $$
Answered by TonyCWX08 last updated on 29/Feb/24
Separate the function:  (√(1−log_(2x+5) ((x+1)^2 )))  1  log_(2x+5) ((x+1)^2 )  2x+5  (x+1)^2   x+1    Domain (√(1−log_(2x+5) ((x+1)^2 )))  x∈⟨−(5/2),−2⟩∪⟨−2,2]  Domain 1  x∈R  Domain log_(2x+5) ((x+1)^2 )  x∈⟨−(5/2),−2⟩∪⟨−2,−1⟩∪⟨−1,∞⟩  Domain 2x+5  x∈R  Domain (x+1)^2   x∈R  Domain x+1  x∈R    Finding Intersection  Domain  x∈⟨−(5/2),−2⟩∪⟨−2,2]
$${Separate}\:{the}\:{function}: \\ $$$$\sqrt{\mathrm{1}−{log}_{\mathrm{2}{x}+\mathrm{5}} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)} \\ $$$$\mathrm{1} \\ $$$${log}_{\mathrm{2}{x}+\mathrm{5}} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right) \\ $$$$\mathrm{2}{x}+\mathrm{5} \\ $$$$\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$${x}+\mathrm{1} \\ $$$$ \\ $$$${Domain}\:\sqrt{\mathrm{1}−{log}_{\mathrm{2}{x}+\mathrm{5}} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right)} \\ $$$$\left.{x}\in\langle−\frac{\mathrm{5}}{\mathrm{2}},−\mathrm{2}\rangle\cup\langle−\mathrm{2},\mathrm{2}\right] \\ $$$${Domain}\:\mathrm{1} \\ $$$${x}\in{R} \\ $$$${Domain}\:{log}_{\mathrm{2}{x}+\mathrm{5}} \left(\left({x}+\mathrm{1}\right)^{\mathrm{2}} \right) \\ $$$${x}\in\langle−\frac{\mathrm{5}}{\mathrm{2}},−\mathrm{2}\rangle\cup\langle−\mathrm{2},−\mathrm{1}\rangle\cup\langle−\mathrm{1},\infty\rangle \\ $$$${Domain}\:\mathrm{2}{x}+\mathrm{5} \\ $$$${x}\in{R} \\ $$$${Domain}\:\left({x}+\mathrm{1}\right)^{\mathrm{2}} \\ $$$${x}\in{R} \\ $$$${Domain}\:{x}+\mathrm{1} \\ $$$${x}\in{R} \\ $$$$ \\ $$$${Finding}\:{Intersection} \\ $$$${Domain} \\ $$$$\left.{x}\in\langle−\frac{\mathrm{5}}{\mathrm{2}},−\mathrm{2}\rangle\cup\langle−\mathrm{2},\mathrm{2}\right] \\ $$

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