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Question Number 184739 by SLVR last updated on 11/Jan/23
Number of linear functions   be defined f:[−1, 1]→[0,2] is  a)1    b)2    c)3   d)4
$${Number}\:{of}\:{linear}\:{functions}\: \\ $$$${be}\:{defined}\:{f}:\left[−\mathrm{1},\:\mathrm{1}\right]\rightarrow\left[\mathrm{0},\mathrm{2}\right]\:{is} \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left.\:\:\:\:{b}\right)\mathrm{2}\:\:\:\:{c}\right)\mathrm{3}\:\:\:{d}\right)\mathrm{4} \\ $$
Commented by SLVR last updated on 11/Jan/23
kindly help me..answer says   2 only..how
$${kindly}\:{help}\:{me}..{answer}\:{says} \\ $$$$\:\mathrm{2}\:{only}..{how} \\ $$
Answered by mr W last updated on 11/Jan/23
in domain [−1,1] the linear function  f(x)=ax+b has the range  [−a+b,a+b] if a>0 (case 1)  or  [a+b,−a+b] if a<0 (case 2)  since the range should be [0,2],  case 1:  −a+b=0  a+b=2  ⇒a=1, b=1 ⇒f(x)=x+1  case 2:  a+b=0  −a+b=2  ⇒a=−1, b=1 ⇒ f(x)=−x+1  i.e. there are 2 functions satisfying  the given conditions:  f(x)=x+1 or f(x)=−x+1
$${in}\:{domain}\:\left[−\mathrm{1},\mathrm{1}\right]\:{the}\:{linear}\:{function} \\ $$$${f}\left({x}\right)={ax}+{b}\:{has}\:{the}\:{range} \\ $$$$\left[−{a}+{b},{a}+{b}\right]\:{if}\:{a}>\mathrm{0}\:\left({case}\:\mathrm{1}\right) \\ $$$${or} \\ $$$$\left[{a}+{b},−{a}+{b}\right]\:{if}\:{a}<\mathrm{0}\:\left({case}\:\mathrm{2}\right) \\ $$$${since}\:{the}\:{range}\:{should}\:{be}\:\left[\mathrm{0},\mathrm{2}\right], \\ $$$${case}\:\mathrm{1}: \\ $$$$−{a}+{b}=\mathrm{0} \\ $$$${a}+{b}=\mathrm{2} \\ $$$$\Rightarrow{a}=\mathrm{1},\:{b}=\mathrm{1}\:\Rightarrow{f}\left({x}\right)={x}+\mathrm{1} \\ $$$${case}\:\mathrm{2}: \\ $$$${a}+{b}=\mathrm{0} \\ $$$$−{a}+{b}=\mathrm{2} \\ $$$$\Rightarrow{a}=−\mathrm{1},\:{b}=\mathrm{1}\:\Rightarrow\:{f}\left({x}\right)=−{x}+\mathrm{1} \\ $$$${i}.{e}.\:{there}\:{are}\:\mathrm{2}\:{functions}\:{satisfying} \\ $$$${the}\:{given}\:{conditions}: \\ $$$${f}\left({x}\right)={x}+\mathrm{1}\:{or}\:{f}\left({x}\right)=−{x}+\mathrm{1} \\ $$
Commented by SLVR last updated on 11/Jan/23
So great of you sir...  so kind of you sir..Thanks again
$${So}\:{great}\:{of}\:{you}\:{sir}… \\ $$$${so}\:{kind}\:{of}\:{you}\:{sir}..{Thanks}\:{again} \\ $$

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