2. Fungsi f: R→R dan g: R→R didefinisikan oleh f(x)=2x+11 dan g(x)=1−x. Tentukan: c. (g∘f−1)(x)

Pertanyaan

2. Fungsi $f:\;\mathbb{R}\rightarrow\mathbb{R}\;\mathrm{dan}\;g:\;\mathbb{R}\rightarrow\mathbb{R}$ didefinisikan oleh $f\left(x\right)=2x+11$ dan $g\left(x\right)=1-x$ . Tentukan:

c. $\left(g\circ f^{-1}\right)\left(x\right)$

R. Diah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

$open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator 13 minus x over denominator 2 end fraction$ . space

Pembahasan

Perhatikan penghitungan berikut!

$f open parentheses x close parentheses equals 2 x plus 11 misal colon space y equals f open parentheses x close parentheses space maka space x equals f to the power of negative 1 end exponent open parentheses x close parentheses f open parentheses x close parentheses equals 2 x plus 11 y equals 2 x plus 11 y minus 11 equals 2 x fraction numerator y minus 11 over denominator 2 end fraction equals x f to the power of negative 1 end exponent open parentheses x close parentheses equals fraction numerator x minus 11 over denominator 2 end fraction g open parentheses x close parentheses equals 1 minus x misal colon space z equals g open parentheses x close parentheses space maka space x equals g to the power of negative 1 end exponent open parentheses x close parentheses g open parentheses x close parentheses equals 1 minus x z equals 1 minus x x equals 1 minus z g to the power of negative 1 end exponent open parentheses x close parentheses equals 1 minus x$

sehingga, open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses yaitu:

$open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals g open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals g open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals 1 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals 2 over 2 minus open parentheses fraction numerator x minus 11 over denominator 2 end fraction close parentheses open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator 2 minus x plus 11 over denominator 2 end fraction open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator 13 minus x over denominator 2 end fraction$

Jadi, $open parentheses g ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses equals fraction numerator 13 minus x over denominator 2 end fraction$ . space