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Diketahui f(x+2)=4x−9 dan (g∘f)(x)=4x−161​, x=4. Tentukan: a. rumus fungsi f b. rumus fungsi g  c. rumus (f∘g)(x)

Pertanyaan

Diketahui f open parentheses x plus 2 close parentheses equals 4 x minus 9 dan open parentheses g ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 1 over denominator 4 x minus 16 end fractionx not equal to 4. Tentukan:

a. rumus fungsi f

b. rumus fungsi g 

c. rumus open parentheses f ring operator g close parentheses open parentheses x close parentheses 

Pembahasan Soal:

a. Misalkan y equals x plus 2 maka diperoleh x equals y minus 2 kemudian substitusikan maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x plus 2 close parentheses end cell equals cell 4 x minus 9 end cell row cell f open parentheses y close parentheses end cell equals cell 4 open parentheses y minus 2 close parentheses minus 9 end cell row blank equals cell 4 y minus 8 minus 9 end cell row blank equals cell 4 y minus 17 end cell row cell f open parentheses y close parentheses end cell equals cell 4 y minus 17 end cell row cell f open parentheses x close parentheses end cell equals cell 4 x minus 17 end cell end table 

Jadi rumus fungsi f adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 17 end table

b. Ingat konsep komposisi fungsi:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell 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 end table

maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell row cell g open parentheses f open parentheses x close parentheses close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell row cell g open parentheses 4 x minus 17 close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell end table

Misalkan y equals 4 x minus 17 maka diperoleh x equals fraction numerator y plus 17 over denominator 4 end fraction sehingga:

table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses 4 x minus 17 close parentheses end cell equals cell fraction numerator 1 over denominator 4 x minus 16 end fraction end cell row cell g open parentheses y close parentheses end cell equals cell fraction numerator 1 over denominator 4 open parentheses begin display style fraction numerator y plus 17 over denominator 4 end fraction end style close parentheses minus 16 end fraction end cell row blank equals cell fraction numerator 1 over denominator y plus 17 minus 16 end fraction end cell row cell g open parentheses y close parentheses end cell equals cell fraction numerator 1 over denominator y plus 1 end fraction end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator 1 over denominator x plus 1 end fraction end cell end table 

Jadi, rumus fungsi g adalah g open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction

c. table attributes columnalign right center left columnspacing 0px end attributes row blank blank f end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 17 end table dan g open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 1 end fraction maka open parentheses f ring operator g close parentheses open parentheses x close parentheses:

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 1 over denominator x plus 1 end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator 1 over denominator x plus 1 end fraction close parentheses minus 17 end cell row blank equals cell fraction numerator 4 over denominator x plus 1 end fraction minus fraction numerator 17 open parentheses x plus 1 close parentheses over denominator x plus 1 end fraction end cell row blank equals cell fraction numerator 4 minus 17 x minus 17 over denominator x plus 1 end fraction end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator negative 17 x minus 13 over denominator x plus 1 end fraction comma space x not equal to negative 1 end cell end table 

Dengan demikian rumus fungsi open parentheses f ring operator g close parentheses open parentheses x close parentheses adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f ring operator g close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator negative 17 x minus 13 over denominator x plus 1 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank not equal to blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Amamah

Mahasiswa/Alumni Universitas Negeri Malang

Terakhir diupdate 06 Oktober 2021

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Pembahasan Soal:

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Diketahui fungsi f:R→R dan g:R→R. Jika diketahui f(3x−4)=5x−1 dan g(x+2)=x2+3x−2. Nilai fungsi (f∘g)(−2) adalah ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell space f left parenthesis 3 x minus 4 right parenthesis end cell equals cell 5 x minus 1 space end cell row blank blank blank row cell 3 x minus 4 end cell equals a row cell 3 x end cell equals cell a plus 4 end cell row x equals cell fraction numerator a plus 4 over denominator 3 end fraction end cell row blank blank blank row cell space f left parenthesis a right parenthesis end cell equals cell 5 open parentheses fraction numerator a plus 4 over denominator 3 end fraction close parentheses minus 1 space end cell row blank equals cell fraction numerator 5 a plus 20 over denominator 3 end fraction minus 1 end cell row blank equals cell fraction numerator 5 a plus 17 over denominator 3 end fraction end cell row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x plus 17 over denominator 3 end fraction end cell row blank blank blank row cell g left parenthesis x plus 2 right parenthesis end cell equals cell x squared plus 3 x minus 2 end cell row blank blank blank row cell x plus 2 end cell equals a row x equals cell a minus 2 end cell row blank blank blank row cell g left parenthesis a right parenthesis end cell equals cell left parenthesis a minus 2 right parenthesis squared plus 3 left parenthesis a minus 2 right parenthesis minus 2 end cell row blank equals cell a squared minus 4 a plus 4 plus 3 a minus 6 minus 2 end cell row blank equals cell a squared minus a minus 4 end cell row cell g left parenthesis x right parenthesis end cell equals cell x squared minus x minus 4 end cell row blank blank blank row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator 5 left parenthesis x squared minus x minus 4 right parenthesis plus 17 over denominator 3 end fraction end cell row blank equals cell fraction numerator 5 x squared minus 5 x minus 3 over denominator 3 end fraction end cell row cell left parenthesis f ring operator g right parenthesis left parenthesis negative 2 right parenthesis end cell equals cell fraction numerator 5 left parenthesis negative 2 right parenthesis squared minus 5 left parenthesis negative 2 right parenthesis minus 3 over denominator 3 end fraction end cell row blank equals cell fraction numerator 20 plus 10 minus 3 over denominator 3 end fraction end cell row blank equals cell 27 over 3 end cell row blank equals 9 end table end style

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Diketahui fungsi f(x)=1+x2​ dan (f∘g)(x)=x−21​x2−4x+5​. Rumus fungsi g(x+3)=...

Pembahasan Soal:

Fungsi komposisi merupakan penggabungan operasi dua jenis fungsi begin mathsize 14px style f left parenthesis x right parenthesis end style dan size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis sehingga menghasilkan sebuah fungsi baru. Operasi fungsi komposisi biasa dilambangkan dengan "size 14px ring operator" dan dibaca komposisi atau bundaran. Fungsi baru yang dapat terbentuk dari begin mathsize 14px style f left parenthesis x right parenthesis end style dan size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis adalah:

a. begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style artinya size 14px g dimasukkan ke begin mathsize 14px style f end style 

b. begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses end style artinya begin mathsize 14px style f end style dimasukkan ke size 14px g 

Sehingga,

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 fraction numerator 1 over denominator x minus 2 end fraction square root of x squared minus 4 x plus 5 end root end cell row cell f open parentheses g open parentheses x close parentheses close parentheses end cell equals cell fraction numerator 1 over denominator x minus 2 end fraction square root of x squared minus 4 x plus 5 end root end cell row cell open parentheses square root of 1 plus open parentheses g open parentheses x close parentheses close parentheses squared end root close parentheses squared end cell equals cell open parentheses fraction numerator 1 over denominator x minus 2 end fraction square root of x squared minus 4 x plus 5 end root close parentheses squared end cell row cell 1 plus open parentheses g open parentheses x close parentheses close parentheses squared end cell equals cell fraction numerator x squared minus 4 x plus 5 over denominator open parentheses x minus 2 close parentheses squared end fraction end cell row cell open parentheses g open parentheses x close parentheses close parentheses squared end cell equals cell fraction numerator x squared minus 4 x plus 5 over denominator open parentheses x minus 2 close parentheses squared end fraction minus 1 end cell row cell open parentheses g open parentheses x close parentheses close parentheses squared end cell equals cell fraction numerator x squared minus 4 x plus 5 minus open parentheses x minus 2 close parentheses squared over denominator open parentheses x minus 2 close parentheses squared end fraction end cell row cell open parentheses g open parentheses x close parentheses close parentheses squared end cell equals cell fraction numerator x squared minus 4 x plus 5 minus x squared plus 4 x minus 4 over denominator open parentheses x minus 2 close parentheses squared end fraction end cell row cell g open parentheses x close parentheses end cell equals cell square root of 1 over open parentheses x minus 2 close parentheses squared end root end cell row cell g open parentheses x close parentheses end cell equals cell fraction numerator 1 over denominator x minus 2 end fraction end cell row blank blank blank end table end style 

Misalkan begin mathsize 14px style h open parentheses x close parentheses equals x plus 3 end style, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator h close parentheses open parentheses x close parentheses end cell equals cell g open parentheses h open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses x plus 3 close parentheses end cell row blank equals cell fraction numerator 1 over denominator x plus 3 minus 2 end fraction end cell row blank equals cell fraction numerator 1 over denominator x plus 1 end fraction end cell end table end style 

Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank g end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses x plus 3 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 1 over denominator x plus 1 end fraction end cell end table end style.

0

Roboguru

Diketahui dan . Maka, adalah ....

Pembahasan Soal:

Pertama, kita cari begin mathsize 14px style straight f open parentheses straight x close parentheses end style dan begin mathsize 14px style straight g open parentheses straight x close parentheses end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f left parenthesis straight x minus 4 right parenthesis end cell equals cell 3 straight x plus 6 space end cell row blank equals cell 3 straight x minus 12 plus 18 space end cell row blank equals cell 3 left parenthesis straight x minus 4 right parenthesis plus 18 space end cell row cell straight f left parenthesis straight x right parenthesis end cell equals cell 3 straight x plus 18 space space end cell row cell straight g left parenthesis 2 straight x minus 4 right parenthesis end cell equals cell negative 4 straight x plus 8 space end cell row blank equals cell negative 2 left parenthesis 2 straight x minus 4 right parenthesis space end cell row cell straight g left parenthesis straight x right parenthesis end cell equals cell negative 2 straight x end cell end table end style

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses end cell equals cell straight f open parentheses straight g open parentheses straight x close parentheses close parentheses end cell row blank equals cell straight f open parentheses negative 2 straight x close parentheses end cell row blank equals cell 3 open parentheses negative 2 straight x close parentheses plus 18 end cell row blank equals cell negative 6 straight x plus 18 end cell end table end style 

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Roboguru

Diketahui dan . Maka, nilai p yang memenuhi dari adalah ....

Pembahasan Soal:

Pertama, kita cari undefined.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight g open parentheses straight x minus 2 close parentheses end cell equals cell 3 minus 2 straight x end cell row blank equals cell negative 1 minus 2 straight x plus 4 end cell row blank equals cell negative 1 minus 2 open parentheses straight x minus 2 close parentheses end cell row cell straight g open parentheses straight x close parentheses end cell equals cell negative 1 minus 2 straight x end cell end table end style 

Sehingga

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight f ring operator straight g close parentheses open parentheses straight x close parentheses end cell equals cell 6 minus 3 straight x end cell row cell straight f open parentheses straight g open parentheses straight x close parentheses close parentheses end cell equals cell 6 minus 3 straight x end cell row cell straight f open parentheses negative 1 minus 2 straight x close parentheses end cell equals cell 6 minus 3 straight x end cell end table end style 

Misalkan begin mathsize 14px style negative 1 minus 2 straight x equals straight a rightwards arrow straight x equals fraction numerator straight a plus 1 over denominator negative 2 end fraction end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses negative 1 minus 2 straight x close parentheses end cell equals cell 6 minus 3 straight x end cell row cell straight f open parentheses straight a close parentheses end cell equals cell 6 minus 3 open parentheses fraction numerator straight a plus 1 over denominator negative 2 end fraction close parentheses end cell row cell straight f open parentheses straight a close parentheses end cell equals cell 6 plus fraction numerator 3 straight a plus 3 over denominator 2 end fraction end cell end table end style 

Diketahui undefined, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses straight p close parentheses end cell equals cell negative 3 end cell row cell 6 plus fraction numerator 3 straight p plus 3 over denominator 2 end fraction end cell equals cell negative 3 end cell row cell fraction numerator 3 straight p plus 3 over denominator 2 end fraction end cell equals cell negative 3 minus 6 end cell row cell fraction numerator 3 straight p plus 3 over denominator 2 end fraction end cell equals cell negative 9 end cell row cell 3 straight p plus 3 end cell equals cell negative 18 end cell row cell 3 straight p end cell equals cell negative 18 minus 3 end cell row cell 3 straight p end cell equals cell negative 21 end cell row straight p equals cell negative 21 over 3 end cell row straight p equals cell negative 7 end cell end table end style

Jadi, jawabannya adalah D.

0

Roboguru

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