Jika \( f(x-1)=x+2 \) dan \( g(x) = \frac{2-x}{x+3} \), maka nilai \( (g^{-1} \circ f )(1) \) adalah…
- \( -6 \)
- \( -2 \)
- \( -\frac{1}{6} \)
- \( \frac{1}{4} \)
- \( 4 \)
Pembahasan:
Untuk menyelesaikan soal ini, kita perlu mencari \(f(x)\) terlebih dahulu. Perhatikan berikut:
\begin{aligned} f(x-1) &= x+2 \\[8pt] f(x-1) &= (x-1)+3 \\[8pt] \text{misalkan} \ x-1 &= a, \ \text{maka} \\[8pt] f(a) &= a+3 \\[8pt] \text{misalkan} \ a &= x, \ \text{maka} \\[8pt] f(x) &= x+3 \end{aligned}
Selanjutnya, kita akan mencari \( g^{-1}(x) \).
\begin{aligned} g(x) = \frac{2-x}{x+3} \Leftrightarrow y &= \frac{2-x}{x+3} \\[8pt] y(x+3) &= 2-x \\[8pt] xy+3y &= 2-x \\[8pt] xy+x &= 2-3y \\[8pt] x(y+1) &= 2-3y \\[8pt] x &= \frac{2-3y}{y+1} \\[8pt] g^{-1}(x) &= \frac{2-3x}{x+1} \end{aligned}
Berdasarkan hasil di atas, kita peroleh berikut ini:
\begin{aligned} (g^{-1} \circ f )(x) &= g^{-1}(f(x)) \\[8pt] &= g^{-1}(x+3) \\[8pt] &= \frac{2-3(x+3)}{(x+3)+1} \\[8pt] &= \frac{2-3x-9}{x+4} \\[8pt] &= \frac{-3x-7}{x+4} \\[8pt] (g^{-1} \circ f )(1) &= \frac{-3 \cdot 1 - 7}{1+4} = -2 \end{aligned}
Jawaban B.