Jika f(x)=4x−1 dan g(x)=2x−3, nilai (f−1∘g)−1(2)adalah...

Pertanyaan

Jika $\style{font-size:14px}{f\left(x\right)=4x-1}$ dan $\style{font-size:14px}{g\left(x\right)=2x-3}$ , nilai $\style{font-size:14px}{\left(f^{-1}\circ g\right)^{-1}\left(2\right)}$ adalah...

Pembahasan:

Ingat kembali sifat invers fungsi komposisi berikut.

$\begin{array}{rcl}\left(f^{-1}\right)^{-1}\left(x\right)&=&f\left(x\right)\\\left(f\circ g\right)^{-1}\left(x\right)&=&\left(f^{-1}\circ g^{-1}\right)\left(x\right)\end{array}$

Dengan menggunakan konsep invers fungsi. Misalkan $begin mathsize 14px style g open parentheses x close parentheses equals y end style$ .

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

Selanjutnya dengan menggunakan sifat invers dari kompisisi fungsi diperoleh

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f to the power of negative 1 end exponent ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell g to the power of negative 1 end exponent open parentheses x close parentheses ring operator open parentheses f to the power of negative 1 end exponent close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses x close parentheses ring operator f open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses 4 x minus 1 close parentheses end cell row blank equals cell fraction numerator 4 x minus 1 plus 3 over denominator 2 end fraction end cell row blank equals cell fraction numerator 4 x plus 2 over denominator 2 end fraction end cell row cell open parentheses f to the power of negative 1 end exponent ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end cell equals cell fraction numerator 4 open parentheses 2 close parentheses plus 2 over denominator 2 end fraction end cell row blank equals cell 10 over 2 end cell row blank equals 5 end table end style$

Dengan demikian nilai dari $begin mathsize 14px style open parentheses f to the power of negative 1 end exponent ring operator g close parentheses to the power of negative 1 end exponent open parentheses 2 close parentheses end style$ adalah $begin mathsize 14px style 5 end style$ .

Jawaban terverifikasi

Dijawab oleh:

Y. Fathoni

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Terakhir diupdate 07 Oktober 2021