Question Number 192732 by mokys last updated on 25/May/23
![let the closed interval [a,b] be the domain of the function f find the domain of f(x−3) and f(x+3) ?](https://www.tinkutara.com/question/Q192732.png)
$${let}\:{the}\:{closed}\:{interval}\:\left[{a},{b}\right]\:{be}\:{the}\:{domain}\:{of}\:{the}\:{function}\:{f}\: \\ $$$${find}\:{the}\:{domain}\:{of}\:{f}\left({x}−\mathrm{3}\right)\:{and}\:{f}\left({x}+\mathrm{3}\right)\:?\:\: \\ $$
Answered by MM42 last updated on 25/May/23
![a≤x−3≤b⇒a+3≤x≤b+3⇒D_(f−3) =[a+3,b+3] a≤x+3≤b⇒a−3≤x≤b−3⇒D_(f+3) =[a−3,b−3]](https://www.tinkutara.com/question/Q192739.png)
$${a}\leqslant{x}−\mathrm{3}\leqslant{b}\Rightarrow{a}+\mathrm{3}\leqslant{x}\leqslant{b}+\mathrm{3}\Rightarrow{D}_{{f}−\mathrm{3}} =\left[{a}+\mathrm{3},{b}+\mathrm{3}\right] \\ $$$${a}\leqslant{x}+\mathrm{3}\leqslant{b}\Rightarrow{a}−\mathrm{3}\leqslant{x}\leqslant{b}−\mathrm{3}\Rightarrow{D}_{{f}+\mathrm{3}} =\left[{a}−\mathrm{3},{b}−\mathrm{3}\right] \\ $$
Answered by Rajpurohith last updated on 26/May/23
$${so}\:{for}\:{a}\leqslant{x}\leqslant{b}\:,{f}\left({x}\right)\:{is}\:{defined} \\ $$$$\Rightarrow{for}\:{a}−\mathrm{3}\leqslant{x}−\mathrm{3}\leqslant{b}−\mathrm{3}\:,{f}\left({x}−\mathrm{3}\right)\:{is}\:{defined}. \\ $$$$\Rightarrow{for}\:{a}+\mathrm{3}\leqslant{x}+\mathrm{3}\leqslant{b}+\mathrm{3}\:,{f}\left({x}+\mathrm{3}\right)\:{is}\:{defined}. \\ $$