Roboguru

Diketahui , dan . Tentukan: a. ,

Pertanyaan

Diketahui f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction comma space g open parentheses x close parentheses equals x squared minus 1, dan h open parentheses x close parentheses equals 3 x plus 5. Tentukan:

a. open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses

Pembahasan Soal:

Ingat kembali:

open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses 

Berarti:

open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals open parentheses h ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses 

Diketahui f open parentheses x close parentheses equals fraction numerator 2 x minus 1 over denominator x plus 1 end fraction comma space g open parentheses x close parentheses equals x squared minus 1, dan h open parentheses x close parentheses equals 3 x plus 5. Maka:

open parentheses h ring operator f close parentheses open parentheses x close parentheses equals h open parentheses f open parentheses x close parentheses close parentheses open parentheses h ring operator f close parentheses open parentheses x close parentheses equals h open parentheses fraction numerator 2 x minus 1 over denominator x plus 1 end fraction close parentheses open parentheses h ring operator f close parentheses open parentheses x close parentheses equals 3 open parentheses fraction numerator 2 x minus 1 over denominator x plus 1 end fraction close parentheses plus 5 open parentheses h ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 6 x minus 2 over denominator x plus 1 end fraction plus 5 open parentheses h ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 6 x minus 2 plus 5 open parentheses x plus 1 close parentheses over denominator x plus 1 end fraction open parentheses h ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 6 x minus 2 plus 5 x plus 5 over denominator x plus 1 end fraction open parentheses h ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 11 x plus 3 over denominator x plus 1 end fraction 

Sehingga:

space space space space open parentheses h ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 11 x plus 3 over denominator x plus 1 end fraction space space space space space space space space space space space space space space space space y equals fraction numerator 11 x plus 3 over denominator x plus 1 end fraction space space space space y times open parentheses x plus 1 close parentheses equals 11 x plus 3 space space space space space space space space space x y plus y equals 11 x plus 3 space space space space space x y minus 11 x equals 3 minus y space space space space x open parentheses y minus 11 close parentheses equals 3 minus y space space space space space space space space space space space space space space space space x equals fraction numerator 3 minus y over denominator y minus 11 end fraction open parentheses h ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 3 minus x over denominator x minus 11 end fraction 

Jadi, open parentheses f to the power of negative 1 end exponent ring operator h to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator 3 minus x over denominator x minus 11 end fraction.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

W. Lestari

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 02 Mei 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

Pertanyaan yang serupa

Fungsi  dan  ditentukan oleh rumus  dan . Tentukan :

Pembahasan Soal:

Pertama kita tentukan invers dari begin mathsize 14px style straight f open parentheses straight x close parentheses equals 7 straight x minus 5 end style 
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight f open parentheses straight x close parentheses end cell equals cell 7 straight x minus 5 end cell row straight y equals cell 7 straight x minus 5 end cell row cell negative 7 straight x end cell equals cell negative straight y minus 5 end cell row straight x equals cell fraction numerator negative straight y minus 5 over denominator negative 7 end fraction end cell row sehingga colon blank row cell straight f to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator negative straight x minus 5 over denominator negative 7 end fraction end cell end table end style 
 

Selanjutnya kita tentukan invers dari  begin mathsize 14px style straight g open parentheses straight x close parentheses equals 4 straight x plus 9 end style
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell straight g open parentheses straight x close parentheses end cell equals cell 4 straight x plus 9 end cell row straight y equals cell 4 straight x plus 9 end cell row cell negative 4 straight x end cell equals cell negative straight y plus 9 end cell row straight x equals cell fraction numerator negative straight y plus 9 over denominator negative 4 end fraction end cell row sehingga colon blank row cell straight g to the power of negative 1 end exponent open parentheses straight x close parentheses end cell equals cell fraction numerator negative straight x plus 9 over denominator negative 4 end fraction end cell end table end style 
 

Maka begin mathsize 14px style open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end style  sebagai berikut :
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses end cell equals cell fraction numerator negative open parentheses straight g to the power of negative 1 end exponent open parentheses straight x close parentheses close parentheses minus 5 over denominator negative 7 end fraction end cell row blank equals cell fraction numerator negative open parentheses begin display style fraction numerator negative straight x plus 9 over denominator negative 4 end fraction end style close parentheses minus 5 over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x minus 9 over denominator negative 4 end fraction end style minus open parentheses begin display style fraction numerator negative 20 over denominator negative 4 end fraction end style close parentheses over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x minus 9 plus 20 over denominator negative 4 end fraction end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator straight x plus 11 over denominator negative 4 end fraction end style over denominator negative 7 end fraction end cell row blank equals cell fraction numerator negative 7 open parentheses straight x plus 11 close parentheses over denominator negative 4 end fraction end cell row blank equals cell fraction numerator negative 7 straight x minus 77 over denominator negative 4 end fraction end cell end table end style 
 

Jadi, begin mathsize 14px style open parentheses straight f to the power of negative 1 end exponent ring operator straight g to the power of negative 1 end exponent close parentheses open parentheses straight x close parentheses equals fraction numerator negative 7 straight x minus 77 over denominator negative 4 end fraction end style 

0

Roboguru

Diketahui fungsi  dan , nilai  jika  adalah

Pembahasan Soal:

Diketahui fungsi f open parentheses x close parentheses equals x cubed dan g open parentheses x close parentheses equals 4 x plus 2, 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 g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses x cubed close parentheses end cell row blank equals cell 4 x cubed plus 2 end cell end table

Inversnya adalah

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 4 x cubed plus 2 end cell row cell 4 x cubed end cell equals cell y minus 2 end cell row cell x cubed end cell equals cell 1 fourth open parentheses y minus 2 close parentheses end cell row x equals cell cube root of 1 fourth open parentheses y minus 2 close parentheses end root end cell row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell cube root of 1 fourth open parentheses x minus 2 close parentheses end root end cell end table

Ingat bahwa open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell row cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell equals cell cube root of 1 fourth open parentheses x minus 2 close parentheses end root end cell row 2 equals cell cube root of 1 fourth open parentheses x minus 2 close parentheses end root end cell row cell 2 cubed end cell equals cell open parentheses cube root of 1 fourth open parentheses x minus 2 close parentheses end root close parentheses cubed end cell row 8 equals cell 1 fourth open parentheses x minus 2 close parentheses end cell row 32 equals cell x minus 2 end cell row x equals cell 32 plus 2 end cell row x equals 34 end table 

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Diketahui fungsi  dan . Jika fungsi    dan , rumus fungsi adalah ...

Pembahasan Soal:

 pertama akan dicari invers dari fungsi begin mathsize 14px style f left parenthesis x right parenthesis equals square root of fraction numerator x plus 2 over denominator 2 x minus 1 end fraction end root space comma space x greater than 1 half comma end style

table attributes columnalign right center left columnspacing 0px end attributes row size 14px y size 14px equals cell square root of fraction numerator size 14px x size 14px plus size 14px 2 over denominator size 14px 2 size 14px x size 14px minus size 14px 1 end fraction end root end cell row cell size 14px y to the power of size 14px 2 end cell size 14px equals cell fraction numerator size 14px x size 14px plus size 14px 2 over denominator size 14px 2 size 14px x size 14px minus size 14px 1 end fraction end cell row cell size 14px 2 size 14px x size 14px y to the power of size 14px 2 size 14px minus size 14px y to the power of size 14px 2 end cell size 14px equals cell size 14px x size 14px plus size 14px 2 end cell row cell size 14px 2 size 14px x size 14px y to the power of size 14px 2 size 14px minus size 14px x end cell size 14px equals cell size 14px y to the power of size 14px 2 size 14px plus size 14px 2 end cell row cell size 14px x begin mathsize 14px style left parenthesis 2 y squared minus 1 right parenthesis end style end cell size 14px equals cell size 14px y to the power of size 14px 2 size 14px plus size 14px 2 end cell row size 14px x size 14px equals cell fraction numerator size 14px y to the power of size 14px 2 size 14px plus size 14px 2 over denominator size 14px 2 size 14px y to the power of size 14px 2 size 14px minus size 14px 1 end fraction end cell row cell size 14px f to the power of size 14px minus size 14px 1 end exponent begin mathsize 14px style left parenthesis x right parenthesis end style end cell size 14px equals cell fraction numerator size 14px x to the power of size 14px 2 size 14px plus size 14px 2 over denominator size 14px 2 size 14px x to the power of size 14px 2 size 14px minus size 14px 1 end fraction end cell end table  

 

selanjutnya akan dicari invers dari fungsi begin mathsize 14px style g left parenthesis x right parenthesis equals fraction numerator x squared over denominator 2 x squared plus 1 end fraction end style

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator x squared over denominator 2 x squared plus 1 end fraction end cell row cell 2 x squared y plus y end cell equals cell x squared end cell row cell 2 x squared y minus x squared end cell equals cell negative y end cell row cell x squared left parenthesis 2 y minus 1 right parenthesis end cell equals cell negative y end cell row cell x squared end cell equals cell fraction numerator negative y over denominator 2 y minus 1 end fraction end cell row x equals cell plus-or-minus square root of fraction numerator negative y over denominator 2 y minus 1 end fraction end root end cell row cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell plus-or-minus square root of fraction numerator negative x over denominator 2 x minus 1 end fraction end root end cell end table  

Dengan menggunakan sifat invers akan ditentukan  begin mathsize 14px style left parenthesis g ring operator f right parenthesis to the power of negative 1 end exponent 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 g ring operator f right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row cell space space space space space space space space space space space space space space space space space space space end cell equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row cell space space space space space space space space space space space space space space space space space space space end cell equals cell f to the power of negative 1 end exponent open parentheses square root of fraction numerator negative x over denominator 2 x minus 1 end fraction end root close parentheses end cell row cell space space space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator open parentheses square root of begin display style fraction numerator negative x over denominator 2 x minus 1 end fraction end style end root close parentheses squared plus 2 over denominator 2 open parentheses square root of fraction numerator negative x over denominator 2 x minus 1 end fraction end root close parentheses squared minus 1 end fraction end cell row cell space space space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator fraction numerator negative x over denominator 2 x minus 1 end fraction plus 2 over denominator 2 open parentheses fraction numerator negative x over denominator 2 x minus 1 end fraction close parentheses minus 1 end fraction end cell row cell space space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator begin display style fraction numerator negative x plus 2 open parentheses 2 x minus 1 close parentheses over denominator 2 x minus 1 end fraction end style over denominator begin display style fraction numerator negative 2 x minus open parentheses 2 x minus 1 close parentheses over denominator 2 x minus 1 end fraction end style end fraction end cell row cell space space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative x plus 4 x minus 2 over denominator up diagonal strike 2 x minus 1 end strike end fraction times fraction numerator up diagonal strike 2 x minus 1 end strike over denominator negative 2 x minus 2 x plus 1 end fraction end cell row cell space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator 3 x minus 2 over denominator negative 4 x plus 1 end fraction end cell row cell space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative open parentheses negative 3 x plus 2 close parentheses over denominator negative open parentheses 4 x minus 1 close parentheses end fraction end cell row cell space space space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 3 x plus 2 over denominator 4 x minus 1 end fraction comma space x not equal to 1 fourth end cell end table end style  

Jadi,jawaban yang tepat adalah C

0

Roboguru

Diketahui . Jika , tentukanlah .

Pembahasan Soal:

Gunakan sifat invers fungsi komposisi dan rumus praktis invers fungsi berikut:

begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses f open parentheses x close parentheses equals a x plus b rightwards double arrow f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus b over denominator a end fraction end style  

Diketahui:
begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x plus 4 over denominator 2 x minus 3 end fraction end style

begin mathsize 14px style g open parentheses x close parentheses equals 3 x plus 2 end style

Akan ditentukan begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style.

Tentukan terlebih dahulu begin mathsize 14px style g to the power of negative 1 end exponent open parentheses x close parentheses end style dengan menggunakan rumus praktis, diperoleh:

begin mathsize 14px style g open parentheses x close parentheses equals 3 x plus 2 rightwards double arrow g to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 2 over denominator 3 end fraction end style 

Sehingga diperoleh begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style dengan perhitungan berikut:

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator x plus 4 over denominator 2 x minus 3 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator x plus 4 over denominator 2 x minus 3 end fraction end style close parentheses minus 2 over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator x plus 4 over denominator 2 x minus 3 end fraction end style minus begin display style fraction numerator 2 open parentheses 2 x minus 3 close parentheses over denominator 2 x minus 3 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator x plus 4 minus 4 x plus 6 over denominator 2 x minus 3 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative 3 x plus 10 over denominator 2 x minus 3 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator negative 3 x plus 10 over denominator 2 x minus 3 end fraction cross times 1 third end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 3 x plus 10 over denominator 6 x minus 9 end fraction end cell end table end style 

Jadi, dipeoleh begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative 3 x plus 10 over denominator 6 x minus 9 end fraction end style.

0

Roboguru

Diketahui  masing-masing adalah fungsi invers dari f dan g. Jika  dan  maka  adalah...

Pembahasan Soal:

Dengan menggunakan sifat fungsi invers maka,

open parentheses f o g close parentheses to the power of negative 1 end exponent space equals space g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent close parentheses space left parenthesis x right parenthesis equals space fraction numerator begin display style fraction numerator 3 minus x over denominator 2 end fraction end style minus 1 over denominator 5 end fraction open parentheses g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent close parentheses space left parenthesis x right parenthesis equals space fraction numerator begin display style fraction numerator 3 minus x over denominator 2 end fraction minus 2 over 2 end style over denominator 5 end fraction open parentheses g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent close parentheses space left parenthesis x right parenthesis equals space fraction numerator begin display style fraction numerator 1 minus x over denominator 2 end fraction end style over denominator 5 end fraction open parentheses g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent close parentheses space left parenthesis x right parenthesis equals space fraction numerator 1 minus x over denominator 10 end fraction open parentheses g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent close parentheses space left parenthesis 6 right parenthesis equals space fraction numerator 1 minus 6 over denominator 10 end fraction open parentheses g to the power of negative 1 end exponent space o space f to the power of negative 1 end exponent close parentheses space left parenthesis 6 right parenthesis equals negative 1 half 

Maka nilainya adalah negative 1 half 

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Ruangguru

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ikuti Kami

©2021 Ruangguru. All Rights Reserved