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zB8NA}lGuj+l4t4GK(%2&F`>^$r~;+GVjptIT9bEut)s7EHdJMp*xhC{ zuGzI^#>+`n7w*N&Ic5|JcI-ipMIlc#PODaSPcPa+y5{)xL;U)QNAA3?qk-!T6!%$U dkC>`e*AF=+VX)+A-24LkueaOjWz=L*{vV~>8#(|0 diff --git a/advanced-math/knowledge/0-perpare/perpare.tex b/advanced-math/knowledge/0-perpare/perpare.tex index f1bf6c3..91d200e 100644 --- a/advanced-math/knowledge/0-perpare/perpare.tex +++ b/advanced-math/knowledge/0-perpare/perpare.tex @@ -53,6 +53,60 @@ 基本初等函数包括:常数函数、幂函数、指数函数、对数函数、三角函数、反三角函数。 +\subparagraph{反函数} \leavevmode \medskip + +反函数的求法: + +\begin{enumerate} + \item 求值域。 + \item 求解。(用$y$表示$x$) + \item 互换$xy$。 +\end{enumerate} + +已知$f(f^{-1}(x))=x$,$f^{-1}(f(x))=x$。为什么是这样?可以验算一下。 + +已知$y=e^x$和$y=\ln x$是一对反函数,$y=\ln e^x=f^{-1}(f(x))=x$。 + +\textbf{例题:}若函数$y=f(x)$的反函数为$y=f^{-1}(x)$,则求$y=f(2x-1)+1$的反函数的解析式。 + +解:整理$y=f(2x-1)+1$,得到$f(2x-1)=y-1$,所以求反函数就是交换$xy$。 + +这里将$2x-1$当作$x$,$y-1$当作$y$,所以得到反函数$2x-1=f^{-1}(y-1)$。 + +所以得到$x=\dfrac{f^{-1}(y-1)+1}{2}$。 + +所以交换表示方法其反函数就是$y=\dfrac{f^{-1}(x-1)+1}{2}$。 + +\textbf{例题:}已知$f(x)=\dfrac{1}{1-x^2}$($x<-1$),求$f^{-1}(-\dfrac{1}{3})$。 + +解:由于是反函数,所以$x$对应$y$,$y$对应$x$。 + +求$f^{-1}(-\dfrac{1}{3})$的值,对应反函数的$x=-\dfrac{1}{3}$,$y=f^{-1}(-\dfrac{1}{3})$的值。 + +即求原函数的$y=-\dfrac{1}{3}$,$x=f(-\dfrac{1}{3})$的值。 + +所以$\dfrac{1}{1-x^2}=-\dfrac{1}{3}$求$x$的值。 + +即$1-x^2=-3$,$x=\pm2$,又$x<-1$,则$x=-2$。 + +\textbf{例题:}已知$f(x)=\dfrac{1-2x}{1+x}$,函数$g(x)$的图像与函数$y=f^{-1}(x+1)$的图像关于$y=x$对称,求$g(5)$。 + +解:由于函数$g(x)$的图像与函数$y=f^{-1}(x+1)$的图像关于$y=x$对称,所以$g(x)$与$f^{-1}(x+1)$也是反函数。 + +所以要求$g(x)$,就要求$f^{-1}(x+1)$。 + +$\because y=f^{-1}(x+1)$,$\therefore x+1=f(y)$,$x=f(y)-1$,即$y=f(x)-1$。 + +$\therefore g(x)=y=f(x)-1$,$g(5)=f(5)-1=-\dfrac{5}{2}$。 + +\textbf{例题:}已知$f(x)=\dfrac{1}{2}(x^2+\sqrt{x+1})$($x\geqslant0$)的反函数为$f^{-1}(x)$,求不等式$f^{-1}(x+1)>3$的解集。 + +解:当$x\geqslant0$时,$f(x)$明显是一个单调递增函数,所以根据反函数性质,其反函数在这个区间上增减性不变也是递增的。 + +$f(0)=\dfrac{1}{2}$,即$f^{-1}(x)$在定义域$[\dfrac{1}{2},+\infty)$上也是递增函数。 + +又$f^{-1}(x+1)>3$,对其求反函数:$f(f^{-1}(x+1))>f(3)$,即$x+1>f(3)$,且$x+1\geqslant\dfrac{1}{2}$,得出$x>\dfrac{9}{2}$。 + \subparagraph{常数函数} \leavevmode \medskip \begin{minipage}{0.35\linewidth} @@ -307,6 +361,8 @@ $\sec x=\dfrac{1}{\cos x},\csc x=\dfrac{1}{\sin x}$: \subparagraph{反三角函数} \leavevmode \medskip +类似是三角函数的反函数,但是由于是个多值函数所以不是严格的函数。所以为了限制反三角函数为单值函数,将反三角函数的值限定在一个区间内,将其作为反三角函数的主值。 + \begin{minipage}{0.45\linewidth} 反正弦函数: @@ -390,7 +446,7 @@ $\sec x=\dfrac{1}{\cos x},\csc x=\dfrac{1}{\sin x}$: \item 单调性:$y=\arctan x$单调增,$y=\textrm{arccot}\,x$单调减。 \item 奇偶性:$y=\arctan x$为奇函数。 \item 有界性:$\vert\arctan x\vert\leqslant\dfrac{\pi}{2}$,$0\leqslant\textrm{arccot}\,x\leqslant\pi$。 - \item 性质:$\arctan x+\textrm{arccot}\,x=\dfrac{\pi}{2}(-\infty