Diketahui fungsi f:R→R, g:R→R, dan h:R→R. Jika fungsi f(x)=4−3x2, g(x)=xx+2​​, dan (f∘g∘h)−1(x)=3x​. Tentukan fungsi h(x).

Pertanyaan

Diketahui fungsi begin mathsize 14px style f colon R rightwards arrow R comma space g colon R rightwards arrow R end style, dan begin mathsize 14px style h colon R rightwards arrow R. end style Jika fungsi begin mathsize 14px style f open parentheses x close parentheses equals 4 minus 3 x squared comma space g open parentheses x close parentheses equals square root of fraction numerator x plus 2 over denominator x end fraction end root comma end style dan begin mathsize 14px style open parentheses f ring operator g ring operator h close parentheses to the power of negative 1 end exponent open parentheses x close parentheses equals square root of 3 x end root. end style Tentukan fungsi begin mathsize 14px style h open parentheses x close parentheses. end style 

H. Firmansyah

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

diperoleh nilai table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell fraction numerator negative 18 over denominator open parentheses x squared minus 12 plus 9 close parentheses end fraction end cell end table.

Pembahasan

Diketahui

left parenthesis f ring operator g ring operator h right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis equals square root of 3 x end root space dan space y equals square root of 3 x end root maka space y squared equals 3 x space rightwards double arrow x equals y squared over 3

Lebih lanjut, diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g ring operator h right parenthesis left parenthesis x right parenthesis end cell equals cell x squared over 3 end cell row cell f left parenthesis g left parenthesis h left parenthesis x right parenthesis right parenthesis right parenthesis end cell equals cell x squared over 3 end cell row cell f open parentheses square root of fraction numerator h left parenthesis x right parenthesis plus 2 over denominator h left parenthesis x right parenthesis end fraction end root close parentheses end cell equals cell x squared over 3 end cell row cell 4 minus 3 open parentheses square root of fraction numerator h left parenthesis x right parenthesis plus 2 over denominator h left parenthesis x right parenthesis end fraction end root close parentheses squared end cell equals cell x squared over 3 end cell row cell 4 minus fraction numerator 3 h left parenthesis x right parenthesis plus 6 over denominator h left parenthesis x right parenthesis end fraction end cell equals cell x squared over 3 end cell row cell fraction numerator 3 h left parenthesis x right parenthesis plus 6 over denominator h left parenthesis x right parenthesis end fraction end cell equals cell 4 minus x squared over 3 end cell row cell fraction numerator 3 h left parenthesis x right parenthesis plus 6 over denominator h left parenthesis x right parenthesis end fraction end cell equals cell fraction numerator 12 minus x squared over denominator 3 end fraction end cell row cell 9 h left parenthesis x right parenthesis plus 18 end cell equals cell 12 h left parenthesis x right parenthesis minus x squared h left parenthesis x right parenthesis end cell row cell open parentheses x squared minus 12 plus 9 close parentheses h left parenthesis x right parenthesis plus 18 end cell equals 0 row cell open parentheses x squared minus 12 plus 9 close parentheses h left parenthesis x right parenthesis end cell equals cell negative 18 end cell row cell h left parenthesis x right parenthesis end cell equals cell fraction numerator negative 18 over denominator open parentheses x squared minus 12 plus 9 close parentheses end fraction end cell end table

Dengan demikian, diperoleh nilai table attributes columnalign right center left columnspacing 0px end attributes row cell h left parenthesis x right parenthesis end cell equals cell fraction numerator negative 18 over denominator open parentheses x squared minus 12 plus 9 close parentheses end fraction end cell end table.

46

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