1.2-bo'lim : Teskari funksiyalar
4. \(g\left( x \right) = 4{\left( {x - 3} \right)^5} + 21\) uchun teskarisini toping. Ushbu bo'limda muhokama qilinganidek, kompozitsiyadan birini yoki ikkalasini hisoblash orqali teskarisini tekshiring.
Maslahat: Ushbu bo'limda tasvirlangan jarayonni eslang. \(g\left( x \right)\)-ni almashtiring, \(x\) va \(y\)-ni almashtiring, \(y\)-ni hal qiling va nihoyat \(y\)-ni \({g^{ - 1}}\left( x \right)\) bilan almashtiring.
Yechim
\[y = 4{\left( {x - 3} \right)^5} + 21\]
2-qadam
\[x = 4{\left( {y - 3} \right)^5} + 21\]
3-qadam
\[\begin{align*}x - 21 & = 4{\left( {y - 3} \right)^5}\\ \frac{1}{4}\left( {x - 21} \right) & = {\left( {y - 3} \right)^5}\\ \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}} & = y - 3\\ & y = 3 + \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}\hspace{0.25in}\hspace{0.25in}\,\,\,\, \to \hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{{g^{ - 1}}\left( x \right) = 3 + \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}}}\end{align*}\]
Va nihoyat, ishimizni tekshirish uchun \(\left( {g \circ {g^{ - 1}}} \right)\left( x \right)\) yoki \(\left( {{g^{ - 1}} \circ g} \right)\left( x \right)\) hisoblang.
4-qadam
Ikkala kompozitsiya ham bajarilishi mumkin, keling, bu holda \(\left( {g \circ {g^{ - 1}}} \right)\left( x \right)\) ni bajaraylik.
\[\begin{align*}\left( {g \circ {g^{ - 1}}} \right)\left( x \right) & = g\left[ {{g^{ - 1}}\left( x \right)} \right]\\ & = 4{\left( {\left[ {3 + \sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}} \right] - 3} \right)^5} + 21\\ & = 4{\left( {\sqrt[5]{{\frac{1}{4}\left( {x - 21} \right)}}} \right)^5} + 21\\ & = 4\left( {\frac{1}{4}\left( {x - 21} \right)} \right) + 21\\ & = \left( {x - 21} \right) + 21\\ & = x\end{align*}\]Shunday qilib, biz \(x\)-ni kompozitsiyadan chiqardik va shuning uchun biz o'z ishimizni to'g'ri bajarganimizni bilamiz.