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Diketahui fungsi Error converting from MathML to accessible text. dan begin mathsize 14px style g colon space text R→R end text end style dengan begin mathsize 14px style f left parenthesis x right parenthesis equals 3 x minus 2 end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals x squared minus 2 x plus 5 end style. Tentukan begin mathsize 14px style f left parenthesis g left parenthesis x right parenthesis right parenthesis end style dan begin mathsize 14px style g left parenthesis f left parenthesis x right parenthesis right parenthesis end style!

 

Pembahasan Video:

Pembahasan Soal:

Diketahui fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals 3 x minus 2 end style dan begin mathsize 14px style g left parenthesis x right parenthesis equals x squared minus 2 x plus 5 end style.

Fungsi komposisi begin mathsize 14px style f left parenthesis g left parenthesis x right parenthesis right parenthesis end style adalah,

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell equals cell f open parentheses x squared minus 2 x plus 5 close parentheses end cell row blank equals cell 3 open parentheses x squared minus 2 x plus 5 close parentheses minus 2 end cell row blank equals cell 3 x squared minus 6 x plus 15 minus 2 end cell row blank equals cell 3 x squared minus 6 x plus 13 end cell end table 

Fungsi komposisi begin mathsize 14px style g left parenthesis f left parenthesis x right parenthesis right parenthesis end style yaitu,

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses f left parenthesis x right parenthesis close parentheses end cell equals cell g open parentheses 3 x minus 2 close parentheses end cell row blank equals cell open parentheses 3 x minus 2 close parentheses squared minus 2 open parentheses 3 x minus 2 close parentheses plus 5 end cell row blank equals cell 9 x squared minus 12 x plus 4 minus 6 x plus 4 plus 5 end cell row blank equals cell 9 x squared minus 18 x plus 13 end cell end table

jadi fungsi komposisi begin mathsize 14px style f left parenthesis g left parenthesis x right parenthesis right parenthesis end style adalah Error converting from MathML to accessible text. dan fungsi komposisi begin mathsize 14px style g left parenthesis f left parenthesis x right parenthesis right parenthesis end style adalah Error converting from MathML to accessible text.

Pembahasan terverifikasi oleh Roboguru

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D. Entry

Terakhir diupdate 07 Juni 2021

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Diektahui fungsi dan carilah jika diketahui: c.

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g ring operator f end cell equals cell g left parenthesis f left parenthesis x right parenthesis right parenthesis end cell row cell g ring operator f end cell equals cell left parenthesis 2 x minus 1 right parenthesis squared minus 1 end cell row blank equals cell 4 x squared minus 4 x plus 1 minus 1 end cell row blank equals cell 4 x squared minus 4 x end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f ring operator g end cell equals cell f left parenthesis g left parenthesis x right parenthesis right parenthesis end cell row cell f ring operator g end cell equals cell 2 open parentheses x squared minus 1 close parentheses minus 1 end cell row blank equals cell 2 x squared minus 2 minus 1 end cell row blank equals cell 2 x squared minus 3 end cell end table end style

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Roboguru

Jika:   Tentukan .

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end cell equals cell open parentheses f left parenthesis g left parenthesis x right parenthesis close parentheses end cell row blank equals cell 2 left parenthesis x squared minus 4 right parenthesis plus 3 end cell row blank equals cell 2 x squared minus 8 plus 3 end cell row blank equals cell 2 x squared minus 5 end cell end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis ring operator open parentheses f ring operator g close parentheses left parenthesis x right parenthesis end cell equals cell h left parenthesis f left parenthesis g left parenthesis x right parenthesis right parenthesis right parenthesis end cell row blank equals cell 3 left parenthesis 2 x squared minus 5 right parenthesis minus 2 end cell row blank equals cell 6 x squared minus 15 minus 2 end cell row blank equals cell 6 x squared minus 17 end cell end table end style 

0

Roboguru

Diketaui fungsi dengan dan fungsi dengan b. selidiki apakah

Pembahasan Soal:

-Untuk begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style

begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g left parenthesis f open parentheses x close parentheses right parenthesis open parentheses g ring operator f close parentheses open parentheses x close parentheses equals g open parentheses 4 x plus 3 close parentheses open parentheses g ring operator f close parentheses open parentheses x close parentheses equals open parentheses 4 x plus 3 close parentheses minus 1 open parentheses g ring operator f close parentheses open parentheses x close parentheses equals 4 x plus 2 end style 

-Untuk begin mathsize 14px style left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell f open parentheses x minus 1 close parentheses end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 4 open parentheses x minus 1 close parentheses plus 3 end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 4 x minus 4 plus 3 end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell 4 x minus 1 end cell end table end style

Jadi, rumus size 14px left parenthesis size 14px g size 14px ring operator size 14px f size 14px right parenthesis begin mathsize 14px style left parenthesis x right parenthesis end style not equal to begin mathsize 14px style left parenthesis f ring operator g right parenthesis end style begin mathsize 14px style left parenthesis x right parenthesis end style

0

Roboguru

Fungsi  dan  saling invers apabila  dengan  adalah fungsi identitas . Dalam hal ini dilambangkan , yaitu  merupakan fungsi invers dari . Misalkan .  1. Tunjukkan  merupakan invers fungsi . 2. Tunjuk...

Pembahasan Soal:

1. Diketahui begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator a x plus b over denominator c x plus d end fraction end style akan dibuktikan begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator negative d x plus b over denominator c x minus a end fraction end style merupakan invers fungsi begin mathsize 14px style f end style.  Misalkan begin mathsize 14px style f open parentheses x close parentheses equals y end style maka berdasarkan konsep invers begin mathsize 14px style x equals g open parentheses x close parentheses end style sehingga diperoleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell fraction numerator a x plus b over denominator c x plus d end fraction end cell row y equals cell fraction numerator a x plus b over denominator c x plus d end fraction end cell row cell y open parentheses c x plus d close parentheses end cell equals cell a x plus b end cell row cell c x y plus d y end cell equals cell a x plus b end cell row cell c x y minus a x end cell equals cell negative d y plus b end cell row cell x open parentheses c y minus a close parentheses end cell equals cell negative d y plus b end cell row x equals cell fraction numerator negative d y plus b over denominator c y minus a end fraction end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator negative d y plus b over denominator c y minus a end fraction end cell end table end style 

Dengan demikian terbukti bahwa begin mathsize 14px style g open parentheses x close parentheses end style invers dari fungsi begin mathsize 14px style f end style

2. Akan ditunjukkan begin mathsize 14px style f ring operator g equals I end style, berdasarkan konsep komposisi fungsi maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses fraction numerator negative d x plus b over denominator c x minus a end fraction close parentheses end cell row blank equals cell fraction numerator a open parentheses fraction numerator negative d x plus b over denominator c x minus a end fraction close parentheses plus b over denominator c open parentheses fraction numerator negative d x plus b over denominator c x minus a end fraction close parentheses plus d end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative a d x plus a b over denominator c x minus a end fraction end style plus begin display style fraction numerator c b x minus a b over denominator c x minus a end fraction end style over denominator begin display style fraction numerator negative d c x plus c b over denominator c x minus a end fraction end style plus begin display style fraction numerator d c x minus a d over denominator c x minus a end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative a d x plus c b x over denominator c x minus a end fraction end style over denominator begin display style fraction numerator c b minus a d over denominator c x minus a end fraction end style end fraction end cell row blank equals cell fraction numerator negative a d x plus c b x over denominator c b minus a d end fraction end cell row blank equals cell fraction numerator x open parentheses c b minus a d close parentheses over denominator c b minus a d end fraction end cell row blank equals x end table end style

Oleh karena diketahui begin mathsize 14px style I open parentheses x close parentheses equals x end style maka terbukti bahwa begin mathsize 14px style f ring operator g equals I end style.

Akan ditunjukkan begin mathsize 14px style g ring operator f equals I end style berdasarkan konsep komposisi fungsi maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g ring operator f open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses fraction numerator a x plus b over denominator c x plus d end fraction close parentheses end cell row blank equals cell fraction numerator negative d open parentheses fraction numerator a x plus b over denominator c x plus d end fraction close parentheses plus b over denominator c open parentheses fraction numerator a x plus b over denominator c x plus d end fraction close parentheses minus a end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative a d x minus b d over denominator c x plus d end fraction end style plus begin display style fraction numerator b c x plus b d over denominator c x plus d end fraction end style over denominator begin display style fraction numerator a c x plus b c over denominator c x plus d end fraction end style plus begin display style fraction numerator negative a c x minus a d over denominator c x plus d end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative a d x plus b c x over denominator c x plus d end fraction end style over denominator begin display style fraction numerator b c minus a d over denominator c x plus d end fraction end style end fraction end cell row blank equals cell fraction numerator x open parentheses b c minus a d close parentheses over denominator b c minus a d end fraction end cell row blank equals x end table end style

Oleh karena diketahui begin mathsize 14px style I open parentheses x close parentheses equals x end style maka terbukti bahwa undefined

Dengan demikia, terbukti bahwa fungsi begin mathsize 14px style f end style dan begin mathsize 14px style g end style saling invers. 

1

Roboguru

1. Diketahui fungsi  dengan  dan fungsi  dengan .  Tentukan : d.

Pembahasan Soal:

Fungsi komposisi merupakan suatu penggabungan dari operasi pada dua jenis fungsi begin mathsize 14px style straight f space left parenthesis straight x right parenthesis space dan space straight g space left parenthesis straight x right parenthesis space end style sampai bisa menghasilkan fungsi baru.

Operasi fungsi komposisi juga biasa dinotasikan dengan penggunaan huruf atau simbol begin mathsize 14px style " ring operator " end styleyang dibaca sebagai komposisi atau bundaran.

Fungsi baru yang dapat terbentuk dari begin mathsize 14px style straight f space left parenthesis straight x right parenthesis space dan space straight g space left parenthesis straight x right parenthesis space end style, yakni:

begin mathsize 14px style open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses equals straight g space dimasukkan space ke space straight f open parentheses straight g ring operator straight f close parentheses open parentheses straight x close parentheses equals straight f space dimasukkan space ke space straight g end style 

Pada soal diketahui :

 begin mathsize 14px style straight f open parentheses straight x close parentheses space equals space 2 straight x minus 5 end style dan  begin mathsize 14px style straight g space open parentheses straight x close parentheses space equals space 3 straight x plus 4 end style

Ditanyakan : tentukan begin mathsize 14px style left parenthesis straight g ring operator straight f right parenthesis left parenthesis negative 1 right parenthesis end style 

Penyelesaian:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis straight x right parenthesis end cell equals cell straight g left parenthesis straight f left parenthesis straight x right parenthesis right parenthesis end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis straight x right parenthesis end cell equals cell straight g left parenthesis 2 straight x minus 5 right parenthesis end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis straight x right parenthesis end cell equals cell 3 left parenthesis 2 straight x minus 5 right parenthesis plus 4 end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis straight x right parenthesis end cell equals cell 6 straight x minus 15 plus 4 end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis straight x right parenthesis end cell equals cell 6 straight x minus 11 end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis negative 1 right parenthesis end cell equals cell 6 left parenthesis negative 1 right parenthesis minus 11 end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis negative 1 right parenthesis end cell equals cell negative 6 minus 11 end cell row cell left parenthesis straight g ring operator straight f right parenthesis left parenthesis negative 1 right parenthesis end cell equals cell negative 17 end cell end table end style   

 

Jadi, begin mathsize 14px style left parenthesis straight g ring operator straight f right parenthesis left parenthesis negative 1 right parenthesis end style = begin mathsize 14px style negative 17 end style 

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