$\ \\3.记f(x)=f_1(x),f(f(x))=f_2(x),f(f(f(x)))=f_3(x),以此类推.\\求出一个f(x)使得f_n(x)=kx+b\;(k>0,k\ne1).\\\ \\\,$
$\LARGE题解$
$\ \\\ \\1.观察到(x^{\sqrt\mu})^{\sqrt\mu}=x^\mu,故f(x)=x^{\sqrt\mu}满足条件.$
$\ \\\displaystyle2.设f(x)=ax+c,则f(f(x))=a^2x+ac+c.\\令a^2=k,ac+c=b,一组解为a=\sqrt k,c=\frac b{\sqrt k+1}.故f(x)=\sqrt kx+\frac b{\sqrt k+1}满足条件.$
$$\begin{align*} f_1(x)&=ax+c,\\ f_2(x)&=a^2x+(a+1)c,\\ f_3(x)&=a^3x+(a^2+a+1)c,\\ \vdots\\ f_n(x)&=a^nx+(a^{n-1}+a^{n-2}+\cdots+1)c\\ &=a^nx+\frac{a^n-1}{a-1}c. \end{align*}$$
$\displaystyle令a^n=k,\frac{a^n-1}{a-1}c=b,一组解为a=\sqrt[n]k,c=\frac{\sqrt[n]k-1}{k-1}b.故f(x)=\sqrt[n]kx+\frac{\sqrt[n]k-1}{k-1}b满足条件.$
