Jika diketahui \( \displaystyle{ f(x) = \frac{x}{x-1}, x \neq 1 } \) dan \( g(x) = f(x^2+1) \). Tentukan \( g(f(x)) \).
Pembahasan:
Kita modifikasi fungsi \( g(x) \) terlebih dahulu, yakni
\begin{aligned} g(x) &= f(x^2+1) \\[1em] &= \frac{(x^2+1)}{(x^2+1)-1} = \frac{x^2+1}{x^2} \\[1em] &= 1 + \frac{1}{x^2} \end{aligned}
Dengan demikian, kita peroleh
\begin{aligned} g(f(x)) &= 1 + \frac{1}{(f(x))^2} \\[1em] &= 1 + \frac{1}{\left( \frac{x}{x-1} \right)^2} \\[1em] &= 1 + \frac{x^2-2x+1}{x^2} \\[1em] &= 2 - \frac{2}{x} + \frac{1}{x^2} \end{aligned}