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Matematika / Membalik Persamaan
« Tulisan terakhir oleh Roni pada Desember 17, 2024, 09:09:05 AM »
\section{Membalik Persamaan}

Andaikan $f \,:\, \mathbb{R} \to \mathbb{R}$ adalah sebarang  pemetaan yang kontinyu.  Persamaan $y = f(x)$ dapat dibalik dengan menyatakan $x$ dalam $f$ dan $y$, yaitu bahwa $x = f^{-1}(y)$ di mana $f^{-1}(y)$ adalah pra-bayangan dari $y$ terhadap $f$, serta $x$ tidak harus tunggal.

Andaikan kali ini $f, g \,:\, \mathbb{R} \to \mathbb{R}$ adalah sebarang dua buah pemetaan yang kontinyu, serta $\alpha \in \mathbb{R}$ adalah sebarang tetapan.  Pemetaan-pemetaan $f\circ g$, $fg$, $(f, g)$, dan $\alpha f$ didefinisikan sedemikian rupa sehingga
\[ (f\circ g)(x) = f(g(x)), \]
\[ (fg)(x) = f(x)g(x), \]
\[ (f, g)(x) = (f(x), g(x)), \]
serta
\[ (\alpha f)(x) = \alpha f(x) \]
untuk semua $x \in \mathbb{R}$.

Pemetaan $c_1, \operatorname{id} \,:\, \mathbb{R} \to \mathbb{R}$ hendak didefinisikan sedemikian rupa sehingga
\[ c_1(x) = 1 \]
dan
\[ \operatorname{id}(x) = x \]
untuk setiap $x \in \mathbb{R}$.

Andaikan kali ini $f \,:\, \mathbb{R}\times\mathbb{R} \to \mathbb{R}$ adalah sebarang pemetaan dengan dua buah valensi.  Persamaan $z = f(x, y)$ dapat dibalik, misalnya, dengan menyatakan $x$ dalam $f$, $y$, dan $z$.  Untuk membaliknya, akan dilakukan prosedur sebagai berikut.
\[ z = f(x, y). \]
\[ z = f(\operatorname{id}(x), yc_1(x)). \]
\[ z = f((\operatorname{id}, yc_1)(x)). \]
\[ z = [f\circ(\operatorname{id}, yc_1)](x). \]
\[ x = [f\circ(\operatorname{id}, yc_1)]^{-1}(z). \]
Dengan demikian $x$ sudah dinyatakan dalam $f$, $y$, dan $z$ di mana $x$ tidak harus tunggal.  Inilah contoh sederhana dari pembalikan persamaan tersebut.  Pembaca dapat mengembangkan sendiri konsep ini dengan cara membuat kasus yang lebih umum untuk persamaan-persamaan yang mengandung pemetaan kontinyu bervalensi lebih dari dua.
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