Diketahui fungsi f(x)=x+32x−5,x=−3 dan g(x)=x2. Fungsi (f∘g)−1(x) adalah ...

Pertanyaan

Diketahui fungsi $\style{font-size:14px}{f\left(x\right)=\frac{2x-5}{x+3},x\neq-3}$ dan $\style{font-size:14px}{g\left(x\right)=\frac2x}$ . Fungsi $\style{font-size:14px}{\left(f\circ g\right)^{-1}\left(x\right)}$ adalah ...

$begin mathsize 14px style fraction numerator 2 minus 4 x over denominator 3 x plus 5 end fraction comma space x not equal to negative 5 over 3 end style$
$fraction numerator size 14px 2 size 14px x size 14px plus size 14px 4 over denominator size 14px 3 size 14px x size 14px plus size 14px 5 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px minus size 14px 5 over size 14px 3$
$fraction numerator size 14px 4 size 14px minus size 14px 2 size 14px x over denominator size 14px 3 size 14px x size 14px minus size 14px 5 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px 5 over size 14px 3$
$fraction numerator size 14px 4 size 14px minus size 14px 2 size 14px x over denominator size 14px 3 size 14px x size 14px plus size 14px 5 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px minus size 14px 5 over size 14px 3$
$fraction numerator size 14px 3 size 14px x size 14px plus size 14px 5 over denominator size 14px 2 size 14px x size 14px minus size 14px 4 end fraction size 14px comma size 14px space size 14px x size 14px not equal to size 14px 2$

Pembahasan:

Pertama kita tentukan invers dari fungsi $begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 2 x minus 5 over denominator x plus 3 end fraction comma x not equal to negative 3 end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell fraction numerator 2 x minus 5 over denominator x plus 3 end fraction end cell row cell 2 x minus 5 end cell equals cell y x plus 3 y end cell row cell 2 x minus y x end cell equals cell 3 y plus 5 end cell row cell x open parentheses 2 minus y close parentheses end cell equals cell 3 y plus 5 end cell row x equals cell fraction numerator 3 y plus 5 over denominator 2 minus y end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 3 x plus 5 over denominator 2 minus x end fraction end cell end table end style$

selanjutnya akan dicari invers dari fungsi $begin mathsize 14px style g open parentheses x close parentheses equals 2 over x end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 2 over x end cell row cell y x end cell equals 2 row x equals cell 2 over y end cell row cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell 2 over x end cell end table end style$

Dengan menggunakan sifat invers fungsi akan dtentukan $begin mathsize 14px style open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end style$

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis f ring operator g right parenthesis to the power of negative 1 end exponent left parenthesis x right parenthesis end cell equals cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell g to the power of negative 1 end exponent open parentheses fraction numerator 3 x plus 5 over denominator 2 minus x end fraction close parentheses end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator 2 over denominator begin display style fraction numerator 3 x plus 5 over denominator 2 minus x end fraction end style end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell 2 times fraction numerator 2 minus x over denominator 3 x plus 5 end fraction end cell row cell space space space space space space space space space space space space space space space end cell equals cell fraction numerator 4 minus 2 x over denominator 3 x plus 5 end fraction comma space x not equal to negative 5 over 3 end cell end table end style$

Jadi,jawaban yang tepat adalah D

Jawaban terverifikasi

Dijawab oleh:

I. Roy

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 06 Oktober 2021