Jelaskan daerah asal dan daerah hasil dari fungsi f(x)=x2+3x+2x+3​

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Jelaskan daerah asal dan daerah hasil dari fungsi  

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F. Ayudhita

Master Teacher

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Ingat!

Bentuk fungsi rasional/pecahan begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator g left parenthesis x right parenthesis over denominator h left parenthesis x right parenthesis end fraction end style dengan begin mathsize 14px style h left parenthesis x right parenthesis not equal to 0 end style 

Pembuat nol penyebut

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus 3 x plus 2 end cell equals 0 row cell left parenthesis x plus 1 right parenthesis left parenthesis x plus 2 right parenthesis end cell equals 0 end table end style 

begin mathsize 14px style x equals negative 1 end style atau begin mathsize 14px style x equals negative 2 end style 

Sehingga domain fungsinya adalah semua bilangan real begin mathsize 14px style x end style kecuali begin mathsize 14px style negative 1 end style dan begin mathsize 14px style negative 2 end style, ditulis  begin mathsize 14px style D subscript f equals left curly bracket x vertical line x element of R comma x not equal to negative 1 comma x not equal to negative 2 right curly bracket end style 

Daerah hasil fungsi diperoleh dengan memisalkan begin mathsize 14px style y equals f left parenthesis x right parenthesis end style maka 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator x plus 3 over denominator x squared plus 3 x plus 2 end fraction end cell row cell y left parenthesis x squared plus 3 x plus 2 right parenthesis end cell equals cell x plus 3 end cell row cell y x squared plus 3 x y plus 2 y end cell equals cell x plus 3 end cell row cell y x squared plus 3 x y plus 2 y minus x minus 3 end cell equals 0 row cell y x squared plus left parenthesis 3 y minus 1 right parenthesis x plus 2 y minus 3 end cell equals 0 end table end style 

Agar fungsi tersebut terdefinisi maka begin mathsize 14px style D greater or equal than 0 end style diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis 3 y minus 1 right parenthesis squared minus 4. y. left parenthesis 2 y minus 3 right parenthesis end cell greater or equal than 0 row cell 9 y squared minus 6 y plus 1 minus 8 y squared plus 12 y end cell greater or equal than 0 row cell y squared plus 6 y plus 1 end cell greater or equal than 0 end table end style 

faktor dari begin mathsize 14px style y squared plus 6 y plus 1 end style dapat diperoleh dengan cara

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y subscript 1 comma 2 end subscript end cell equals cell fraction numerator negative 6 plus-or-minus square root of 6 squared minus 4.1.1 end root over denominator 2.1 end fraction end cell row blank equals cell fraction numerator negative 6 plus-or-minus square root of 36 minus 4 end root over denominator 2 end fraction end cell row blank equals cell fraction numerator negative 6 plus-or-minus square root of 32 over denominator 2 end fraction end cell row blank equals cell fraction numerator negative 6 plus-or-minus 4 square root of 2 over denominator 2 end fraction end cell row blank equals cell negative 3 plus-or-minus 2 square root of 2 end cell end table end style 

diperoleh begin mathsize 14px style y subscript 1 equals negative 3 plus 2 square root of 2 end style dan begin mathsize 14px style y subscript 2 equals negative 3 minus 2 square root of 2 end style, kemudian dibuat garis bilangan seperti berikut ini

Sehingga daerah hasil fungsi adalah begin mathsize 14px style y less or equal than negative 3 minus 2 square root of 2 end style atau begin mathsize 14px style y greater or equal than negative 3 plus 2 square root of 2 end style 

Jadi, daerah asal fungsi adalah   begin mathsize 14px style bold italic D subscript bold f bold equals begin bold style left curly bracket right enclose x x element of real numbers comma x not equal to negative 1 comma x not equal to negative 2 right curly bracket end style end style dan daerah hasil fungsi adalah  begin mathsize 14px style bold italic R subscript bold f bold equals begin bold style left curly bracket right enclose y y less or equal than negative 3 minus 2 square root of 2 space a t a u space y greater or equal than negative 3 plus 2 square root of 2 comma y element of real numbers right curly bracket end style end style 

 

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