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Diketahui fungsi  dan . Tentukanlah  dan !

Diketahui fungsi begin mathsize 14px style f open parentheses x close parentheses equals 2 x plus 3 end style dan begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x plus 1 close parentheses equals negative 2 x squared minus 4 x minus 1 end style. Tentukanlah begin mathsize 14px style g to the power of negative 1 end exponent open parentheses x close parentheses end style dan begin mathsize 14px style g to the power of negative 1 end exponent open parentheses negative 2 close parentheses end style!

Jawaban:

Diingat salah satu sifat dari invers fungsi kompoposisi yaitu

begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x close parentheses equals f open parentheses g open parentheses x close parentheses close parentheses end style  

begin mathsize 14px style open curly brackets f to the power of negative 1 end exponent ring operator open parentheses f ring operator g close parentheses close curly brackets open parentheses x close parentheses equals g open parentheses x close parentheses end style

Diketahui begin mathsize 14px style open parentheses f ring operator g close parentheses open parentheses x plus 1 close parentheses equals negative 2 x squared minus 4 x minus 1 end style dan begin mathsize 14px style f open parentheses x close parentheses equals 2 x plus 3 end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis left parenthesis x plus 1 right parenthesis end cell equals cell negative 2 x squared minus 4 x minus 1 end cell row blank equals cell negative 2 x squared minus 4 x minus 2 plus 1 end cell row blank equals cell negative 2 open parentheses x squared plus 2 x plus 1 close parentheses plus 1 end cell row blank equals cell negative 2 open parentheses x plus 1 close parentheses squared plus 1 end cell row blank rightwards double arrow cell left parenthesis f ring operator g right parenthesis left parenthesis x right parenthesis equals negative 2 x squared plus 1 end cell end table end style 

Dicari fungsi invers 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 misalkan 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 y end style diperoleh

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell negative 2 x squared plus 1 end cell row blank left right double arrow cell y minus 1 equals negative 2 x squared end cell row blank left right double arrow cell x squared equals fraction numerator y minus 1 over denominator negative 2 end fraction end cell row blank left right double arrow cell x equals square root of fraction numerator y minus 1 over denominator negative 2 end fraction end root comma y less or equal than 1 end cell row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell square root of fraction numerator y minus 1 over denominator negative 2 end fraction end root comma y less or equal than 1 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 square root of fraction numerator x minus 1 over denominator negative 2 end fraction end root comma x less or equal than 1 end cell end table end style 

Karena diketahui 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 open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end style dan begin mathsize 14px style open curly brackets open parentheses f ring operator g close parentheses ring operator g to the power of negative 1 end exponent close curly brackets open parentheses x close parentheses equals f open parentheses x close parentheses end style maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open curly brackets left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent ring operator f close curly brackets left parenthesis x right parenthesis end cell equals cell left curly bracket left parenthesis g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent right parenthesis ring operator f right curly bracket left parenthesis x right parenthesis end cell row blank equals cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell end table end style

Dicari begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis end style menggunakan persamaan di atas.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell left curly bracket left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent ring operator f right curly bracket left parenthesis x right parenthesis end cell row blank equals cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent open parentheses 2 x plus 3 close parentheses end cell row blank equals cell square root of fraction numerator open parentheses 2 x plus 3 close parentheses minus 1 over denominator negative 2 end fraction end root end cell row blank equals cell square root of fraction numerator 2 x plus 2 over denominator negative 2 end fraction end root end cell row blank equals cell square root of negative x minus 1 end root comma x less or equal than negative 1 end cell end table end style  

Dengan demikian diperoleh begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis x right parenthesis equals square root of negative x minus 1 end root comma x less or equal than negative 1 end style.

Selanjutnya dicari begin mathsize 14px style g to the power of negative 1 end exponent open parentheses negative 2 close parentheses end style.

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell g to the power of negative 1 end exponent left parenthesis negative 2 right parenthesis end cell equals cell square root of negative open parentheses negative 2 close parentheses minus 1 end root end cell row blank equals cell square root of 1 end cell row blank equals 1 end table end style 

Dengan demikian diperoleh begin mathsize 14px style g to the power of negative 1 end exponent left parenthesis negative 2 right parenthesis equals 1 end style

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