Pertanyaan

Diketahui fungsi f ( x ) = − x − 8 dan g ( x ) = 3 2 x − 5 dan h ( x ) = 3 x + 4. Jika k ( x ) = ( f − g ) ( x ) , invers dari ( h ∘ k ) ( x ) adalah ...

Diketahui fungsi $f (x) = - x - 8$ dan $g (x) = \frac{2 x - 5}{3}$ dan $h (x) = 3 x + 4.$ Jika $k (x) = (f - g) (x)$ , invers dari $(h \circ k) (x)$ adalah ...

$begin mathsize 14px style fraction numerator negative 5 x minus 19 over denominator 3 end fraction end style$
$begin mathsize 14px style fraction numerator 15 minus x over denominator 5 end fraction end style$
$begin mathsize 14px style fraction numerator 15 minus x over denominator 5 end fraction end style$
$begin mathsize 14px style negative fraction numerator x plus 5 over denominator 5 end fraction end style$
$begin mathsize 14px style negative fraction numerator x plus 15 over denominator 5 end fraction end style$

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E

Pembahasan

Pertama kita tentukan fungsi dengan cara berikut selanjutnya akan dicari invers dari invers dari fungsi ditentukan dengan cara berikut dengan sifat invers komposisi akan ditentukan Jadi, jawaban yang tepat adalah E

Pertama kita tentukan fungsi begin mathsize 14px style k open parentheses x close parentheses end style dengan cara berikut

$begin mathsize 14px style k open parentheses x close parentheses equals open parentheses f minus g close parentheses open parentheses x close parentheses space space space space space space equals open parentheses negative x minus 8 close parentheses minus fraction numerator 2 x minus 5 over denominator 3 end fraction space space space space space space equals fraction numerator 3 open parentheses negative x minus 8 close parentheses minus open parentheses 2 x minus 5 close parentheses over denominator 3 end fraction space space space space space space equals fraction numerator negative 3 x minus 24 minus 2 x plus 5 over denominator 3 end fraction space space space space space space equals fraction numerator negative 5 x minus 19 over denominator 3 end fraction end style$

selanjutnya akan dicari invers dari begin mathsize 14px style k open parentheses x close parentheses end style

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator negative 5 x minus 19 over denominator 3 end fraction end cell row cell 3 y end cell equals cell negative 5 x minus 19 end cell row cell 5 x end cell equals cell negative 19 minus 3 y end cell row x equals cell fraction numerator negative 19 minus 3 y over denominator 5 end fraction end cell row cell k to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator negative 19 minus 3 x over denominator 5 end fraction end cell end table end style$

invers dari fungsi begin mathsize 14px style h open parentheses x close parentheses end style ditentukan dengan cara berikut

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 3 x plus 4 end cell row cell 3 x end cell equals cell y minus 4 end cell row x equals cell fraction numerator y minus 4 over denominator 3 end fraction end cell row cell h to the power of negative 1 end exponent end cell equals cell fraction numerator x minus 4 over denominator 3 end fraction end cell end table end style$

dengan sifat invers komposisi akan ditentukan begin mathsize 14px style open parentheses h ring operator k close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis h ring operator k right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell k to the power of negative 1 end exponent open parentheses h to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell k to the power of negative 1 end exponent open parentheses fraction numerator x minus 4 over denominator 3 end fraction close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 19 minus up diagonal strike 3 begin display style fraction numerator x minus 4 over denominator up diagonal strike 3 end fraction end style over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 19 minus x plus 4 over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator negative 15 minus x over denominator 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell negative fraction numerator x plus 15 over denominator 15 end fraction end cell end table end style$

Jadi, jawaban yang tepat adalah E