Question

determinați a,b stiind ca functia f:[1,4] ->[1,7] f(x)=ax+b este bijectiva

Answer 1

$\displaystyle Functia~f~este~o~functie~de~gradul~ intai ,~prin~urmare~ea~este \\ \\ injectiva.\\ \\ Pentru~a~fi~bijectiva,~este~necesar~sa~fie~si~surjectiva. \\ \\ Deci~pentru~orice~y \in [1;7],~ecuatia~f(x)=y~trebuie~sa~aiba~o \\ \\ solutie.~(Evident,~x \in [1;4])\\ \\ Insa~orice~functie~de~gradul~intai~este~fie~strict~crescatoare, \\ \\ fie~strict~descrescatoare,~ceea~ce~inseamna~ca~exista~2~cazuri:$

$\displaystyle i)~\left \{ {{f(1)=1} \atop {f(4)=7}} \right. ~si~ii)~ \left \{ {{f(1)=7} \atop {f(4)=1}} \right. . \\ \\ i)~ \Leftrightarrow \left \{ {{a+b=1} \atop {4a+b=7}} \right. . \\ \\ (4a+b)-(a+b)=7-1 \Leftrightarrow 3a=6 \Rightarrow a=2. \\ \\ a+b=1 \Rightarrow b=1-a=1-2=-1. \\ \\ Deci~am~gasit~functia~f(x)=2x-1. \\ \\ ii) \Leftrightarrow \left \{ {{a+b=7} \atop {4a+b=1}} \right. . \\ \\ (4a+b)-(a+b)=1-7 \Leftrightarrow 3a=-6 \Rightarrow a=-2. \\ \\ a+b=7 \Rightarrow b=7-a=7-(-2)=9.$

$\displaystyle Deci~am~gasit~functia~f(x)=-2x+9. \\ \\ Avem~doua~perechi~de~solutii:~(a,b) \in \left \{(2,-1);(-2;9) \right \}.$