1. Given the function $f(f(f(f(f(x)))))=16 x+15$, find the analytical expression of the linear function $f(x)$
1. Let a linear function $f(x)=a x+b(a \neq 1)$, then from $a^{4}\left(x-\frac{b}{1-a}\right)+\frac{b}{1-a}=16 x+15$, we have $\left\{\begin{array}{l}a^{4}=16, \\ \frac{b\left(1-a^{4}\right)}{1-a}=15\end{array} \Rightarrow\left\{\begin{array}{l}a=2 \\ b=1\end{array}\right.\right.$ or $\left\{\begin{array}{l}a=-2, \\ b=-3 .\end{array}\right.$
Therefore, the required linear functions are $f(x)=2 x+1$ or $f(x)=-2 x+3$.