3.7-bo'lim : Teskari funksiyalar
2. Berilgan \(\displaystyle g\left( x \right) = \frac{1}{2}x + 7\) \({g^{ - 1}}\left( x \right)\) ni toping.
Yechim
Birinchi qadam uchun funksiyani \(y\) bilan almashtiramiz.
\[y = \frac{1}{2}x + 7\]2-qadam
Keyin barcha \(x\)-larni \(y\)-ga va barcha original \(y\)-larni \(x\)-ga almashtiring.
\[x = \frac{1}{2}y + 7\]3-qadam
\(y\) uchun 2-bosqichdagi tenglamani yeching.
\[\begin{align*}x & = \frac{1}{2}y + 7\\ 2x & = y + 14\\ 2x - 14 & = y\end{align*}\]4-qadam
\(y\) o'rniga \({g^{ - 1}}\left( x \right)\).
\[{g^{ - 1}}\left( x \right) = 2x - 14\]5-qadam
Nihoyat, \(\left( {g \circ {g^{ - 1}}} \right)\left( x \right)\) va \(\left( {{g^{ - 1}} \circ g} \right)\left( x \right)\) dan birini yoki ikkalasini hisoblash orqali tezkor tekshirib ko'ring va ularning har biri \(x\) ekanligini tekshiring. Umuman olganda, biz odatda ulardan birini tekshiramiz va buni bu erda qilamiz.
\[\left( {g \circ {g^{ - 1}}} \right)\left( x \right) = g\left[ {{g^{ - 1}}\left( x \right)} \right] = g\left[ {2x - 14} \right] = \frac{1}{2}\left( {2x - 14} \right) + 7 = x - 7 + 7 = x\]Tekshirish amalga oshiriladi, shuning uchun biz ishni to'g'ri bajarganimizni va teskarisini bilamiz.