How to find $f(x)$ in the composite function $f(f(x)) = 6x-8$ where f(x) is a linear function

Question

Hi just want to know how to solve this.

find $f(x)$ in the composite function $f\bigl(f(x)\bigr) = 6x-8$

For more of this see: https://math.stackexchange.com/questions/2254594/how-much-does-f-circ-f-determine-f/2256219 https://en.wikipedia.org/wiki/Functional_square_root — user357980, Jun 20 '17 at 06:14
Question posted with no context, irrelevant answer posted 7 minutes later and accepted 9 minutes after it was posted: MSE at its "best"... — Did, Jun 20 '17 at 06:27
This question is from an exam and does not have any context, the question was by itself in the exam. How is there a need for a context? If context was needed Archis wouldn't have been able to solve it. I accepted it because Archis gave pretty much what the solution manual gave and his explanation is sound. — shawn, Jun 20 '17 at 06:48
@Did. It is obvious (at least to me) that things are changing in this (potentially) fantastic site. — Claude Leibovici, Jun 20 '17 at 06:52

score 2 · Answer 1 · answered Jun 20 '17 at 06:14

2

If you consider a linear function $f(x)=mx+b$, we have $f(f(x))=m(mx+b)+b$.

This is equal to $m^2x+mb+b$.

We have that $m^2=6$, so $m=\sqrt{6}$.

Therefore, $\sqrt{6}b+b=8$,

We have $b(\sqrt{6}+1)=8$, so $b=\displaystyle \frac{8}{\sqrt{6}-1}=\frac{8\sqrt{6}+8}{5}$

Therefore, $\displaystyle f(x)=x\sqrt{6}+\frac{8\sqrt{6}+8}{5}$, is a solution.

answered Jun 20 '17 at 06:14

Saketh Malyala

there may be others, but this is the most readily derivable – Saketh Malyala Jun 20 '17 at 06:14

score 1 · Answer 2 · answered Jun 20 '17 at 06:13

1

Such $f$ is not uniquely determined, but we might try an ansatz $f(x)=ax+b$. Then $f(f(x))=a^2x+ab+b$. So solve the equations $a^2=6$, $(a+1)b=-8$.

answered Jun 20 '17 at 06:13

Hagen von Eitzen

2 Answers2