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Diketahui invers fungsi komposisi  dan fungsi . Ma...

Diketahui invers fungsi komposisi open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 2 x minus 1 over denominator x plus 3 end fraction dan fungsi f left parenthesis x right parenthesis equals 5 x minus 8. Maka nilai g left parenthesis 1 right parenthesis adalah...

Jawaban:

Misalkan open parentheses f ring operator g close parentheses to the power of negative 1 end exponent left parenthesis x right parenthesis equals y, Maka

 table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 2 x minus 1 over denominator x plus 3 end fraction end cell row cell y open parentheses x plus 3 close parentheses end cell equals cell 2 x minus 1 end cell row cell x y plus 3 y end cell equals cell 2 x minus 1 end cell row cell x y minus 2 x end cell equals cell negative 1 minus 3 y end cell row cell x open parentheses y minus 2 close parentheses end cell equals cell negative 1 minus 3 y end cell row x equals cell fraction numerator negative 1 minus 3 y over denominator y minus 2 end fraction end cell end table 

Karena x equals open parentheses f o g close parentheses left parenthesis y right parenthesis, maka left parenthesis f o g right parenthesis left parenthesis y right parenthesis equals fraction numerator negative 1 minus 3 y over denominator y minus 2 end fraction. Sehingga left parenthesis f o g right parenthesis left parenthesis x right parenthesis equals fraction numerator negative 1 minus 3 x over denominator x minus 2 end fraction.

Pada soal, diketahui fungsi f left parenthesis x right parenthesis equals 5 x minus 8, ditanyakan nilai g left parenthesis x right parenthesis di x equals 1 atau g left parenthesis 1 right parenthesis.

Penyelesaian:

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator negative 1 minus 3 x over denominator x minus 2 end fraction end cell row cell f open parentheses g left parenthesis x right parenthesis close parentheses end cell equals cell fraction numerator negative 1 minus 3 x over denominator x minus 2 end fraction end cell row cell 5 open parentheses g left parenthesis x right parenthesis close parentheses minus 8 end cell equals cell fraction numerator negative 1 minus 3 x over denominator x minus 2 end fraction end cell row cell 5 open parentheses g left parenthesis x right parenthesis close parentheses end cell equals cell fraction numerator negative 1 minus 3 x over denominator x minus 2 end fraction plus 8 end cell row cell g left parenthesis x right parenthesis end cell equals cell 1 fifth open parentheses fraction numerator negative 1 minus 3 x over denominator x minus 2 end fraction plus 8 close parentheses end cell row cell g left parenthesis x right parenthesis end cell equals cell 1 fifth open parentheses fraction numerator negative 1 minus 3 x plus 8 open parentheses x minus 2 close parentheses over denominator x minus 2 end fraction close parentheses end cell row cell g left parenthesis x right parenthesis end cell equals cell 1 fifth open parentheses fraction numerator negative 1 minus 3 x plus 8 x minus 16 over denominator x minus 2 end fraction close parentheses end cell row cell g left parenthesis x right parenthesis end cell equals cell 1 fifth open parentheses fraction numerator 5 x minus 17 over denominator x minus 2 end fraction close parentheses end cell row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 17 over denominator 5 x minus 10 end fraction end cell end table 

Nilai g left parenthesis x right parenthesis di x equals 1 yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell g left parenthesis x right parenthesis end cell equals cell fraction numerator 5 x minus 17 over denominator 5 x minus 10 end fraction end cell row cell g left parenthesis 1 right parenthesis end cell equals cell fraction numerator 5 left parenthesis 1 right parenthesis minus 17 over denominator 5 left parenthesis 1 right parenthesis minus 10 end fraction end cell row cell g left parenthesis 1 right parenthesis end cell equals cell fraction numerator negative 12 over denominator negative 5 end fraction end cell row cell g left parenthesis 1 right parenthesis end cell equals cell 12 over 5 end cell end table 


Dengan demikian, nilai dari g left parenthesis 1 right parenthesis adalah 12 over 5.

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