1.2-bo'lim : Teskari funksiyalar
1. \(f\left( x \right) = 6x + 15\) uchun teskarisini toping. Ushbu bo'limda muhokama qilinganidek, kompozitsiyadan birini yoki ikkalasini hisoblash orqali teskarisini tekshiring.
Maslahat: Ushbu bo'limda tasvirlangan jarayonni eslang. \(f\left( x \right)\)-ni almashtiring, \(x\) va \(y\)-ni almashtiring, \(y\)-ni hal qiling va nihoyat \(y\)-ni \({f^{ - 1}}\left( x \right)\) bilan almashtiring.
Yechim
\[y = 6x + 15\]
2-qadam
\[x = 6y + 15\]
3-qadam
\[\begin{align*}x - 15 & = 6y\\ y & = \frac{1}{6}\left( {x - 15} \right)\hspace{0.25in}\hspace{0.25in}\,\,\,\, \to \hspace{0.25in}\hspace{0.25in}\require{bbox} \bbox[2pt,border:1px solid black]{{{f^{ - 1}}\left( x \right) = \frac{1}{6}\left( {x - 15} \right)}}\end{align*}\]
Va nihoyat, ishimizni tekshirish uchun \(\left( {f \circ {f^{ - 1}}} \right)\left( x \right)\) yoki \(\left( {{f^{ - 1}} \circ f} \right)\left( x \right)\) hisoblang.
3-qadam
Ikkala kompozitsiya ham bajarilishi mumkin, keling, bu holda \(\left( {f \circ {f^{ - 1}}} \right)\left( x \right)\) ni bajaraylik.
\[\begin{align*}\left( {f \circ {f^{ - 1}}} \right)\left( x \right) & = f\left[ {{f^{ - 1}}\left( x \right)} \right]\\ & = 6\left[ {\frac{1}{6}\left( {x - 15} \right)} \right] + 15\\ & = x - 15 + 15\\ & = x\end{align*}\]Shunday qilib, biz \(x\)-ni kompozitsiyadan chiqardik va shuning uchun biz o'z ishimizni to'g'ri bajarganimizni bilamiz.