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Perhatikan grafik fungsi kuadrat berikut. Berdasarkan grafik tersebut, nilai f ( 2 ) + f − 1 ( − 3 ) adalah ...

Perhatikan grafik fungsi kuadrat berikut.

Berdasarkan grafik tersebut, nilai  adalah ...

  1. ...space 

  2. ...undefined 

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S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

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Pembahasan

Perhatikan gambar grafik pada soal! Diketahui: Dua titik yang memotong sumbu- adalah dan . Maka dan . Titik yang memotong sumbu- adalah . Maka . Persamaan kuadrat Cari terlebih dahulu nilai seperti berikut: Maka persamaan kuadrat adalah: Substitusi ke persamaan kuadrat seperti berikut: Diketahui: Fungsi invers dari fungsi kuadrat adalah . , dengan , dan . Maka, Substitusi ke fungsi invers seperti berikut: Nilai adalah

Perhatikan gambar grafik pada soal!

Diketahui:

Dua titik yang memotong sumbu-begin mathsize 14px style x end style adalah begin mathsize 14px style open parentheses negative 1 comma space 0 close parentheses end style dan begin mathsize 14px style open parentheses 4 comma space 0 close parentheses end style. Maka begin mathsize 14px style open parentheses x subscript 1 comma space y subscript 1 close parentheses equals open parentheses negative 1 comma space 0 close parentheses end style dan begin mathsize 14px style open parentheses x subscript 2 comma space y subscript 2 close parentheses equals open parentheses 4 comma space 0 close parentheses end style.

Titik yang memotong sumbu-begin mathsize 14px style y end style adalah begin mathsize 14px style open parentheses 0 comma space 2 close parentheses end style. Maka begin mathsize 14px style open parentheses x comma space y close parentheses equals open parentheses 0 comma space 2 close parentheses end style.

Persamaan kuadrat begin mathsize 14px style y equals a open parentheses x minus x subscript 1 close parentheses open parentheses x minus x subscript 2 close parentheses end style

Cari terlebih dahulu nilai begin mathsize 14px style a end style seperti berikut:

begin mathsize 14px style table attributes columnalign right center left end attributes row y equals cell a open parentheses x minus x subscript 1 close parentheses open parentheses x minus x subscript 1 close parentheses end cell row 2 equals cell a open parentheses 0 minus open parentheses negative 1 close parentheses close parentheses open parentheses 0 minus 4 close parentheses end cell row 2 equals cell a open parentheses 1 close parentheses open parentheses negative 4 close parentheses end cell row 2 equals cell negative 4 a end cell row cell fraction numerator 2 over denominator negative 4 end fraction end cell equals a row cell negative 1 half end cell equals a end table end style

Maka persamaan kuadrat adalah:

begin mathsize 14px style table attributes columnalign center center left end attributes row cell f open parentheses x close parentheses equals y end cell equals cell a open parentheses x minus x subscript 1 close parentheses open parentheses x minus x subscript 2 close parentheses end cell row cell f open parentheses x close parentheses equals y end cell equals cell negative 1 half open parentheses x minus open parentheses negative 1 close parentheses close parentheses open parentheses x minus 4 close parentheses end cell row space equals cell negative 1 half open parentheses x plus 1 close parentheses open parentheses x minus 4 close parentheses end cell row space equals cell negative 1 half open parentheses x squared minus 3 x minus 4 close parentheses end cell row space equals cell negative 1 half x squared plus 3 over 2 x plus 2 end cell end table end style 

Substitusi begin mathsize 14px style f open parentheses 2 close parentheses end style ke persamaan kuadrat begin mathsize 14px style f open parentheses x close parentheses end style seperti berikut:

begin mathsize 14px style table attributes columnalign center center left end attributes row cell f open parentheses x close parentheses end cell equals cell negative 1 half x squared plus 3 over 2 x plus 2 end cell row cell f open parentheses 2 close parentheses end cell equals cell negative 1 half open parentheses 2 close parentheses squared plus fraction numerator 3 over denominator down diagonal strike 2 end fraction open parentheses down diagonal strike 2 close parentheses plus 2 end cell row space equals cell negative 1 half open parentheses 4 close parentheses plus 3 plus 2 end cell row space equals cell negative 2 plus 3 plus 2 end cell row space equals 3 end table end style

Diketahui:

Fungsi invers dari fungsi kuadrat begin mathsize 14px style f open parentheses x close parentheses equals a x squared plus b x plus c end style adalah begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals negative fraction numerator b over denominator 2 a end fraction minus square root of fraction numerator b squared minus 4 a open parentheses c minus x close parentheses over denominator 4 a squared end fraction end root end style.

begin mathsize 14px style f open parentheses x close parentheses equals negative 1 half x squared plus 3 over 2 x plus 2 end style, dengan begin mathsize 14px style a equals negative 1 half end stylebegin mathsize 14px style b equals 3 over 2 end style dan begin mathsize 14px style c equals 2 end style.

Maka,

 begin mathsize 14px style table attributes columnalign right right left end attributes row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell negative fraction numerator b over denominator 2 a end fraction minus square root of fraction numerator b squared minus 4 a open parentheses c minus x close parentheses over denominator 4 a squared end fraction end root end cell row space equals cell negative fraction numerator begin display style 3 over 2 end style over denominator 2 open parentheses negative begin display style 1 half end style close parentheses end fraction minus square root of fraction numerator open parentheses 3 over 2 close parentheses squared minus 4 open parentheses negative begin display style 1 half end style close parentheses open parentheses 2 minus x close parentheses over denominator 4 open parentheses negative begin display style 1 half end style close parentheses squared end fraction end root end cell row space equals cell negative fraction numerator begin display style 3 over 2 end style over denominator negative 1 end fraction minus square root of fraction numerator begin display style 9 over 4 end style plus 2 open parentheses 2 minus x close parentheses over denominator down diagonal strike 4 open parentheses begin display style fraction numerator 1 over denominator down diagonal strike 4 end fraction end style close parentheses end fraction end root end cell row space equals cell 3 over 2 minus square root of 9 over 4 plus 4 minus 2 x end root end cell row space equals cell 3 over 2 minus square root of fraction numerator 25 minus 8 x over denominator 4 end fraction end root end cell row space equals cell 3 over 2 minus fraction numerator square root of 25 minus 8 x end root over denominator 2 end fraction end cell row space equals cell fraction numerator 3 minus square root of 25 minus 8 x end root over denominator 2 end fraction end cell end table end style 

Substitusi begin mathsize 14px style f open parentheses negative 3 close parentheses end style ke fungsi invers begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses end style seperti berikut:

begin mathsize 14px style table attributes columnalign right center left end attributes row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 3 minus square root of 25 minus 8 x end root over denominator 2 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses negative 3 close parentheses end cell equals cell fraction numerator 3 minus square root of 25 minus 8 open parentheses negative 3 close parentheses end root over denominator 2 end fraction end cell row space equals cell fraction numerator 3 minus square root of 25 plus 24 end root over denominator 2 end fraction end cell row space equals cell fraction numerator 3 minus square root of 49 over denominator 2 end fraction end cell row space equals cell fraction numerator 3 minus 7 over denominator 2 end fraction end cell row space equals cell fraction numerator negative 4 over denominator 2 end fraction end cell row space equals cell negative 2 end cell end table end style 

Nilai begin mathsize 14px style f open parentheses 2 close parentheses plus f to the power of negative 1 end exponent open parentheses negative 3 close parentheses end style adalah

begin mathsize 14px style table attributes columnalign center center left end attributes row cell f left parenthesis 2 right parenthesis plus f to the power of negative 1 end exponent left parenthesis negative 3 right parenthesis end cell equals cell 3 plus left parenthesis negative 2 right parenthesis end cell row space equals cell 3 minus 2 end cell row space cell space equals end cell cell 1 space end cell end table end style   

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