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Diketahui fungsi f ( x ) = 3 x + 4 2 x − 5 ​ dan g ( x ) = 5 x − 3 6 x + 2 ​ . Tentukan invers fungsi komposisi ( f ∘ g ) − 1 ( x ) .

Diketahui fungsi  dan . Tentukan invers fungsi komposisi 

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I. Kumaralalita

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

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invers fungsi komposisi adalah .

 invers fungsi komposisi begin mathsize 14px style open parentheses f text ∘ end text g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end style adalah begin mathsize 14px style fraction numerator 6 x plus 19 over denominator 38 x plus 13 end fraction end style.

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Pembahasan

Bentuk umum dari invers fungsi rasional adalah . Fungsi invers dari adalah . Fungsi invers dari adalah . Maka invers fungsi komposisi dapat dihitung seperti berikut, Jadi,invers fungsi komposisi adalah .

Bentuk umum dari invers fungsi rasional begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator a x plus b over denominator c x plus d end fraction end style adalah begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative d x plus b over denominator c x minus a end fraction end style.

Fungsi invers dari begin mathsize 14px style f left parenthesis x right parenthesis equals fraction numerator 2 x minus 5 over denominator 3 x plus 4 end fraction end style adalah begin mathsize 14px style f to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 4 x minus 5 over denominator 3 x minus 2 end fraction end style.

Fungsi invers dari size 14px g size 14px left parenthesis size 14px x size 14px right parenthesis size 14px equals fraction numerator size 14px 6 size 14px x size 14px plus size 14px 2 over denominator size 14px 5 size 14px x size 14px minus size 14px 3 end fraction adalah begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 3 x plus 2 over denominator 5 x minus 6 end fraction end style.

Maka invers fungsi komposisi begin mathsize 14px style open parentheses f text ∘ end text g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end style dapat dihitung seperti berikut,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end cell row blank 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 left parenthesis x right parenthesis end cell row blank blank blank row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator negative 4 x minus 5 over denominator 3 x minus 2 end fraction close parentheses end cell row blank blank blank row blank equals cell fraction numerator 3 open parentheses begin display style fraction numerator negative 4 x minus 5 over denominator 3 x minus 2 end fraction end style close parentheses plus 2 over denominator 5 open parentheses begin display style fraction numerator negative 4 x minus 5 over denominator 3 x minus 2 end fraction end style close parentheses minus 6 end fraction end cell row blank blank blank row blank equals cell fraction numerator begin display style fraction numerator 3 left parenthesis negative 4 x minus 5 right parenthesis plus 2 left parenthesis 3 x minus 2 right parenthesis over denominator up diagonal strike 3 x minus 2 end strike end fraction end style over denominator begin display style fraction numerator 5 left parenthesis negative 4 x minus 5 right parenthesis minus 6 left parenthesis 3 x minus 2 right parenthesis over denominator up diagonal strike 3 x minus 2 end strike end fraction end style end fraction end cell row blank blank blank row blank equals cell fraction numerator negative 12 x minus 15 plus 6 x minus 4 over denominator negative 20 x minus 25 minus 18 x plus 12 end fraction end cell row blank blank blank row blank equals cell fraction numerator negative 6 x minus 19 over denominator negative 38 x minus 13 end fraction times open parentheses fraction numerator negative 1 over denominator negative 1 end fraction close parentheses end cell row blank blank blank row blank equals cell fraction numerator 6 x plus 19 over denominator 38 x plus 13 end fraction end cell end table end style      

Jadi, invers fungsi komposisi begin mathsize 14px style open parentheses f text ∘ end text g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis end style adalah begin mathsize 14px style fraction numerator 6 x plus 19 over denominator 38 x plus 13 end fraction end style.

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