Roboguru

Diketahui fungsi f:R→R dan g:R→R. Jika fungsi  f(x)=2x−1x+2​​,x>21​,  dan g(x)=2x2+1x2​, rumus fungsi (g∘f)−1(x)adalah ...

Pertanyaan

Diketahui fungsi begin mathsize 14px style f colon R rightwards arrow R end style dan begin mathsize 14px style g colon R rightwards arrow R end style. Jika 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  dan 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, rumus fungsi 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 styleadalah ... 

  1. begin mathsize 14px style fraction numerator 3 x plus 2 over denominator 4 x minus 1 end fraction comma space x not equal to 1 fourth end style    

  2. fraction numerator size 14px 3 size 14px x size 14px minus size 14px 2 over denominator size 14px 4 size 14px x size 14px minus size 14px 1 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px 1 over size 14px 4  

  3. fraction numerator size 14px minus size 14px 3 size 14px x size 14px plus size 14px 2 over denominator size 14px 4 size 14px x size 14px minus size 14px 1 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px 1 over size 14px 4   

  4. begin mathsize 14px style fraction numerator negative 3 x plus 2 over denominator 4 x minus 5 end fraction comma space x not equal to 5 over 4 end style   

  5. begin mathsize 14px style fraction numerator negative 2 x plus 3 over denominator 4 x minus 1 end fraction comma space x not equal to 1 fourth end style   

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

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Roy

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 06 Oktober 2021

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