Relation between matrices and a function composed with itself

Question

I was looking through an older notebook and I saw this problem that I dont understand, it says if$$f(x) =\frac{3x+1}{1-x},x\neq1 $$Compute $$F=\underbrace{f \circ \dots \ \circ f(4)}_{2016\:\text{times} }$$ I have this as a solution. Consider: $$A=\begin{pmatrix} 3&1\\-1&1\end{pmatrix}$$ then $$A\begin{pmatrix} x\\1\end{pmatrix}= \begin{pmatrix} 3x+1\\-x+1\end{pmatrix}$$ and $$F=A^n \begin{pmatrix} x\\1\end{pmatrix}$$ It's indeed easy to prove by induction after computing a few powers of A that $$A^n=2^{n-1}\begin{pmatrix}n+2&n \\-n&-n+2\end{pmatrix}$$now taking $x=4$ and multiplying both sides by $\begin{pmatrix} x\\1\end{pmatrix} $ we have that$$A^n\begin{pmatrix} 4\\1\end{pmatrix}=2^{n-1}\begin{pmatrix}5n+8\\-5n+2\end{pmatrix} $$So for $n=2016$ we have $$F=\frac{5\cdot2016+8}{-5\cdot2016+2}=-\frac{5044}{5039}$$ Now, to be clear I don't understand why putting the coefficients of a function into a matrix would not deteriorize my answer, or why does this work? Can we use the same method to higher powers of $x$? and what if we had for example $$f(x)=(3x+1)(1-x)$$ what matrix should be considered? I would love to get clarifications if possible, many thanks in advance!

Answer 1

There is an explanation of the link between functions of the form $\frac{ax+b}{cx+d}$ and invertible 2x2 matrices here. Mapping a function like this to a 2x2 matrix is useful because composition of functions corresponds to multiplication of matrices.

Answer 2

HINT COMMENT.- You can always do that which is simply a multiplication of a matrix $2\text{x}2$ by a matrix $2\text{x}1$ getting as product a matrix $2\text{x}1$. The problem is to find out a convenient matrix. On the other hand I guess it would be possible to find a general expression for $f^{(n)}(x)$ without usage of matrices however it like difficult enough. You have
$$f^{(2)}(x)=\frac{2x+1}{-x}\\f^{(3)}(x)=\frac{5x+3}{-3x-1}\\f^{(4)}(x)=\frac{9x+5}{4x+2}$$ I don't see an easy way to use induction (maybe a trick for successive iterations).......