Roboguru

Nilai fungsi invers  dari .

Pertanyaan

Nilai fungsi invers begin mathsize 14px style straight f to the power of negative 1 end exponent left parenthesis 2 right parenthesis end style dari begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals fraction numerator 3 straight x plus 4 over denominator 2 straight x minus 1 end fraction comma space straight x not equal to 1 half space end style.

Pembahasan Soal:

Diketahui:

begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals fraction numerator 3 straight x plus 4 over denominator 2 straight x minus 1 end fraction comma space straight x not equal to 1 half space end style

Dengan menerapkan konsep invers fungsi,

Misalkan begin mathsize 14px style straight y equals straight f left parenthesis straight x right parenthesis rightwards arrow straight x equals straight f to the power of negative 1 end exponent left parenthesis straight y right parenthesis end style sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight f left parenthesis straight x right parenthesis end cell equals cell fraction numerator 3 straight x plus 4 over denominator 2 straight x minus 1 end fraction end cell row straight y equals cell fraction numerator 3 straight x plus 4 over denominator 2 straight x minus 1 end fraction end cell row cell straight y left parenthesis 2 straight x minus 1 right parenthesis end cell equals cell 3 straight x plus 4 end cell row cell 2 yx minus straight y end cell equals cell 3 straight x plus 4 end cell row cell 2 yx minus 3 straight x end cell equals cell straight y plus 4 end cell row cell straight x left parenthesis 2 straight y minus 3 right parenthesis end cell equals cell straight y plus 4 end cell row straight x equals cell fraction numerator straight y plus 4 over denominator left parenthesis 2 straight y minus 3 right parenthesis end fraction end cell row cell straight f to the power of negative 1 end exponent left parenthesis straight y right parenthesis end cell equals cell fraction numerator straight y plus 4 over denominator left parenthesis 2 straight y minus 3 right parenthesis end fraction end cell row cell straight f to the power of negative 1 end exponent left parenthesis 2 right parenthesis end cell equals cell fraction numerator 2 plus 4 over denominator 2 times 2 minus 3 end fraction end cell row blank equals cell fraction numerator 6 over denominator 4 minus 3 end fraction end cell row blank equals cell 6 over 1 end cell row blank equals 6 end table end style 

Jadi, Jika begin mathsize 14px style straight f left parenthesis straight x right parenthesis equals fraction numerator 3 straight x plus 4 over denominator 2 straight x minus 1 end fraction comma space straight x not equal to 1 half space end style maka begin mathsize 14px style straight f to the power of negative 1 end exponent left parenthesis 2 right parenthesis equals 6 end style.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

R. RGFLLIMA

Terakhir diupdate 30 April 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

Jika  adalah invers dari , maka nilai

Pembahasan Soal:

Jika begin mathsize 14px style g to the power of negative 1 end exponent end style adalah invers dari begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator 8 minus 3 x over denominator 4 minus x end fraction comma space x not equal to 4 end style, maka nilai begin mathsize 14px style g to the power of negative 1 end exponent open parentheses 4 close parentheses equals... end style 

begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator 8 minus 3 x over denominator 4 minus x end fraction space space space space y equals fraction numerator 8 minus 3 x over denominator 4 minus x end fraction y open parentheses 4 minus x close parentheses equals 8 minus 3 x 4 y minus x y equals 8 minus 3 x 3 x minus x y equals 8 minus 4 y x open parentheses 3 minus y close parentheses equals 8 minus 4 y x equals fraction numerator 8 minus 4 y over denominator 3 minus y end fraction g to the power of negative 1 end exponent open parentheses y close parentheses equals fraction numerator 8 minus 4 y over denominator 3 minus y end fraction g to the power of negative 1 end exponent open parentheses 4 close parentheses equals fraction numerator 8 minus 4 open parentheses 4 close parentheses over denominator 3 minus 4 end fraction space space space space space space space space space space space space equals 8 end style 

 

Roboguru

Jika , maka tentukan .

Pembahasan Soal:

Menentukan f to the power of negative 1 end exponent left parenthesis x right parenthesis

table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x minus 5 over denominator x minus 9 end fraction end cell row y equals cell fraction numerator 2 x minus 5 over denominator x minus 9 end fraction end cell row cell y left parenthesis x minus 9 right parenthesis end cell equals cell 2 x minus 5 end cell row cell x y minus 9 y end cell equals cell 2 x minus 5 end cell row cell x y minus 2 x end cell equals cell 9 y minus 5 end cell row cell x left parenthesis y minus 2 right parenthesis end cell equals cell 9 y minus 5 end cell row x equals cell fraction numerator 9 y minus 5 over denominator y minus 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis y right parenthesis end cell equals cell fraction numerator 9 y minus 5 over denominator y minus 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator 9 x minus 5 over denominator x minus 2 end fraction end cell end table 

f to the power of negative 1 end exponent left parenthesis x right parenthesis akan memiliki nilai jika penyebutnya tidak sama dengan nol, sehingga

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus 2 end cell not equal to 0 row x not equal to 2 end table 

Jadi, invers dari f left parenthesis x right parenthesis adalah f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 9 x minus 5 over denominator x minus 2 end fraction comma space x not equal to 2.

Roboguru

Invers dari fungsi  adalah

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis x right parenthesis end cell equals cell fraction numerator 2 x minus 7 over denominator 3 x plus 8 end fraction end cell row y equals cell fraction numerator 2 x minus 7 over denominator 3 x plus 8 end fraction end cell row cell 3 x y plus 8 y end cell equals cell 2 x minus 7 end cell row cell 3 x y minus 2 x end cell equals cell negative 8 y minus 7 end cell row cell x left parenthesis 3 y minus 2 right parenthesis end cell equals cell negative 8 y minus 7 end cell row x equals cell fraction numerator negative 8 y minus 7 over denominator 3 y minus 2 end fraction end cell row cell f to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell fraction numerator negative 8 x minus 7 over denominator 3 x minus 2 end fraction end cell end table end style

Roboguru

Jika , maka tentukan .

Pembahasan Soal:

misal space space fraction numerator 2 x minus 5 over denominator x minus 9 end fraction equals a space space space space space space space space space space space space 2 x minus 5 equals a open parentheses x minus 9 close parentheses space space space space space space space space space space space space 2 x minus 5 equals a x minus 9 a space space space space space space space space space 2 x minus a x equals negative 9 a plus 5 space space space space space space space space space open parentheses 2 minus a close parentheses x equals negative 9 a plus 5 space space space space space space space space space space space space space space space space space space space space x equals fraction numerator negative 9 a plus 5 over denominator 2 minus a end fraction space space space space space space space space space space space space space space space space space space space space x equals fraction numerator 9 a minus 5 over denominator a minus 2 end fraction 

Jadi, f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator 9 x minus 5 over denominator x minus 2 end fraction comma space x not equal to 2.

Roboguru

Fungsi  ditentukan oleh  dan  adalah invers fungsi , maka  ...

Pembahasan Soal:

Tentukan terlebih dahulu fungsi f open parentheses x close parentheses.

Misal:

x plus 2 equals a, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 2 end cell equals a row x equals cell a minus 2 end cell end table

Substitusi x equals a minus 2 ke fungsi f open parentheses x plus 2 close parentheses equals fraction numerator x minus 2 over denominator x plus 4 end fraction.

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x plus 2 close parentheses end cell equals cell fraction numerator x minus 2 over denominator x plus 4 end fraction end cell row cell f open parentheses a close parentheses end cell equals cell fraction numerator a minus 2 minus 2 over denominator a minus 2 plus 4 end fraction end cell row cell f open parentheses a close parentheses end cell equals cell fraction numerator a minus 4 over denominator a plus 2 end fraction end cell row cell f open parentheses x close parentheses end cell equals cell fraction numerator x minus 4 over denominator x plus 2 end fraction end cell end table

Kemdian tentukan invers fungsi f open parentheses x close parentheses.

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell fraction numerator x minus 4 over denominator x plus 2 end fraction end cell row y equals cell fraction numerator x minus 4 over denominator x plus 2 end fraction end cell row cell y open parentheses x plus 2 close parentheses end cell equals cell x minus 4 end cell row cell x y plus 2 y end cell equals cell x minus 4 end cell row cell x y minus x end cell equals cell negative 2 y minus 4 end cell row cell x open parentheses y minus 1 close parentheses end cell equals cell negative 2 y minus 4 end cell row x equals cell fraction numerator negative 2 y minus 4 over denominator y minus 1 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 2 x minus 4 over denominator x minus 1 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 2 x plus 4 over denominator negative x plus 1 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 2 x plus 4 over denominator 1 minus x end fraction comma space x not equal to 1 end cell end table

Maka f to the power of negative 1 end exponent open parentheses x close parentheses equalsfraction numerator 2 x plus 4 over denominator 1 minus x end fraction semicolon space x not equal to 1.

Jadi, jawaban yang tepat adalah A.

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