Pertanyaan

Ditentukan fungsi f : R → R , g : R → R , dan h : R → R dengan f ( x ) = x + 4 1 , g ( x ) = 3 x , dan h ( x ) = x − 1 . Rumus ( h ∘ g ∘ f ) − 1 ( 1 − x ) = ... .

Ditentukan fungsi $f : R \to R$ , $g : R \to R$ , dan $h : R \to R$ dengan $f (x) = \frac{1}{x + 4}$ , $g (x) = 3 x$ , dan $h (x) = x - 1$ . Rumus $(h \circ g \circ f)^{- 1} (1 - x) = ...$ .

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

Y. Fathoni

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta.

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah B.

Pembahasan

Gunakan konsep komposisi fungsi dan invers fungsi. Diketahui: Akan ditentukan . Terlebih dahulu tentukan , diperoleh: Kemudian tentukan , perhatikan perhitungan berikut: Sehingga dapat diperoleh sebagai berikut: Jadi, jawaban yang tepat adalah B.

Gunakan konsep komposisi fungsi dan invers fungsi.

Diketahui:

$begin mathsize 14px style f open parentheses x close parentheses equals fraction numerator 1 over denominator x plus 4 end fraction end style$

Akan ditentukan .

Terlebih dahulu tentukan , diperoleh:

$begin mathsize 14px style table attributes columnalign right center left columnspacing 2px end attributes row cell left parenthesis h ring operator g ring operator f right parenthesis open parentheses x close parentheses end cell equals cell h open parentheses open parentheses g ring operator f close parentheses open parentheses x close parentheses close parentheses end cell row blank equals cell h open parentheses g open parentheses fraction numerator 1 over denominator x plus 4 end fraction close parentheses close parentheses end cell row blank equals cell h open parentheses 3 open parentheses fraction numerator 1 over denominator x plus 4 end fraction close parentheses close parentheses end cell row blank equals cell h open parentheses fraction numerator 3 over denominator x plus 4 end fraction close parentheses end cell row blank equals cell open parentheses fraction numerator 3 over denominator x plus 4 end fraction close parentheses minus 1 end cell row blank equals cell fraction numerator 3 over denominator x plus 4 end fraction minus fraction numerator x plus 4 over denominator x plus 4 end fraction end cell row blank equals cell fraction numerator 3 minus x minus 4 over denominator x plus 4 end fraction end cell row cell left parenthesis h ring operator g ring operator f right parenthesis left parenthesis x right parenthesis end cell equals cell fraction numerator negative x minus 1 over denominator x plus 4 end fraction end cell end table end style$

Kemudian tentukan , perhatikan perhitungan berikut:

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

Sehingga dapat diperoleh sebagai berikut:

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

Jadi, jawaban yang tepat adalah B.