Diketahui dan . Invers fungsi adalah ....

Pertanyaan

Diketahui $begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to 2 end style$ dan $begin mathsize 14px style g open parentheses x close parentheses equals 6 x minus 2 end style$ . Invers fungsi $begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style$ adalah ....

$begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator negative 13 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end style$
$begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator negative 13 over denominator 6 x plus 30 end fraction comma space x not equal to negative 5 end style$
$begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 13 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end style$
$begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 14 over denominator 6 x plus 30 end fraction comma space x not equal to negative 5 space end style$
$begin mathsize 14px style left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals fraction numerator 14 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end style$

Pembahasan Soal:

Ada 3 langkah untuk menentukan fungsi invers, yaitu:

Ubahlah bentuk $begin mathsize 14px style y equals f open parentheses x close parentheses end style$ menjadi bentuk $begin mathsize 14px style x equals f open parentheses y close parentheses end style$ .
Tuliskan $begin mathsize 14px style x end style$ sebagai $begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses end style$ sehingga $begin mathsize 14px style f to the power of negative 1 end exponent open parentheses y close parentheses equals f open parentheses y close parentheses end style$ .
Ubahlah variabel $begin mathsize 14px style y end style$ dengan $begin mathsize 14px style x end style$ sehingga diperoleh rumus fungsi invers $begin mathsize 14px style f to the power of negative 1 end exponent open parentheses x close parentheses end style$ .

Diketahui:
$begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 5 x minus 3 over denominator x plus 2 end fraction comma space x not equal to 2 end style$
$begin mathsize 14px style g open parentheses x close parentheses equals 6 x minus 2 end style$
Akan ditentukan invers fungsi $begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style$ .

Tentukan $begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style$ terlebih dahulu, diperoleh:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell f open parentheses g open parentheses x close parentheses close parentheses end cell row blank equals cell f open parentheses 6 x minus 2 close parentheses end cell row blank equals cell fraction numerator 5 open parentheses 6 x minus 2 close parentheses minus 3 over denominator open parentheses 6 x minus 2 close parentheses plus 2 end fraction end cell row blank equals cell fraction numerator 30 x minus 10 minus 3 over denominator 6 x minus 2 plus 2 end fraction end cell row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses end cell equals cell fraction numerator 30 x minus 13 over denominator 6 x end fraction end cell end table end style$

Sehingga invers fungsi $begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style$ atau $begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style$ dapat dihitung sebagai berikut:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell open parentheses f ring operator g close parentheses open parentheses x close parentheses equals y end cell equals cell fraction numerator 30 x minus 13 over denominator 6 x end fraction end cell row cell y times 6 x end cell equals cell 30 x minus 13 end cell row cell 6 x y end cell equals cell 30 x minus 13 end cell row cell 6 x y minus 30 x end cell equals cell negative 13 end cell row cell x open parentheses 6 y minus 30 close parentheses end cell equals cell negative 13 end cell row cell x equals open parentheses f ring operator g close parentheses open parentheses y close parentheses end cell equals cell fraction numerator negative 13 over denominator 6 y minus 30 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 13 over denominator 6 x minus 30 end fraction comma space x not equal to 5 end cell end table end style$

Jadi, jawaban yang tepat adalah A.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 13 Maret 2021

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