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Diketahui fungsi  dan . Nilai dari  adalah ....

Diketahui fungsi begin mathsize 14px style f open parentheses x close parentheses equals 4 x minus 3 end style dan begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator x minus 1 over denominator 2 x plus 4 end fraction end style. Nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 5 close parentheses end style adalah ....

Jawaban:

Gunakan konsep fungsi komposisi dan invers fungsi komposisi.

Diketahui fungsi begin mathsize 14px style f open parentheses x close parentheses equals 4 x minus 3 end style dan begin mathsize 14px style g open parentheses x close parentheses equals fraction numerator x minus 1 over denominator 2 x plus 4 end fraction end style.

Akan ditentukan nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 5 close parentheses end style.

Terlebih dahulu tentukan begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses end style.

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 fraction numerator x minus 1 over denominator 2 x plus 4 end fraction close parentheses end cell row blank equals cell 4 open parentheses fraction numerator x minus 1 over denominator 2 x plus 4 end fraction close parentheses minus 3 end cell row blank equals cell fraction numerator 4 x minus 4 over denominator 2 x plus 4 end fraction minus 3 end cell row blank equals cell fraction numerator 4 x minus 4 minus 3 open parentheses 2 x plus 4 close parentheses over denominator 2 x plus 4 end fraction end cell row blank equals cell fraction numerator 4 x minus 4 minus 6 x minus 12 over denominator 2 x plus 4 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 negative 2 x minus 16 over denominator 2 x plus 4 end fraction end cell end table end style 

Kemudian tentukan 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.

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 negative 2 x minus 16 over denominator 2 x plus 4 end fraction end cell row cell y open parentheses 2 x plus 4 close parentheses end cell equals cell negative 2 x minus 16 end cell row cell 2 x y plus 4 y end cell equals cell negative 2 x minus 16 end cell row cell 2 x y plus 2 x end cell equals cell negative 4 y minus 16 end cell row cell x open parentheses 2 y plus 2 close parentheses end cell equals cell negative 4 y minus 16 end cell row x equals cell fraction numerator negative 4 y minus 16 over denominator 2 y plus 2 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 4 x minus 16 over denominator 2 x plus 2 end fraction end cell end table end style 

Kemudian tentukan nilai dari begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 5 close parentheses end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes 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 4 x minus 16 over denominator 2 x plus 2 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 5 close parentheses end cell equals cell fraction numerator negative 4 open parentheses 5 close parentheses minus 16 over denominator 2 open parentheses 5 close parentheses plus 2 end fraction end cell row blank equals cell fraction numerator negative 20 minus 16 over denominator 10 plus 2 end fraction end cell row blank equals cell fraction numerator negative 36 over denominator 12 end fraction end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 5 close parentheses end cell equals cell negative 3 end cell end table end style 

Jadi, diperoleh begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses 5 close parentheses equals negative 3 end style.

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