Trouble finding the Linear Functions that satisfy the given Identity

[SolXero]

Homework Statement

Given f[f(x)] = 2x + 1 find all linear functions that satisfy this identity.
Given f[f[f(x)]] = 2x + 1 find all linear functions that satisfy this identity.


2. The attempt at a solution

I have not started to attempt a solution at this because I have no idea how to do these types of problems.

 

[Mark44]

Homework Statement

Given f[f(x)] = 2x + 1 find all linear functions that satisfy this identity.
Given f[f[f(x)]] = 2x + 1 find all linear functions that satisfy this identity.


2. The attempt at a solution

I have not started to attempt a solution at this because I have no idea how to do these types of problems.

You're give that f(x) is a linear function, so let f(x) = ax + b.
For the first problem, you know that f(f(x)) = 2x + 1, so f(ax + b) = 2x + 1, or a(ax + b) + b = 2x + 1. Solve for a and b. Hint: a and b are not rational numbers.

Solve the second problem in a similar manner.

 

[SolXero]

Thanks a lot!

For the first problem the answer would be:

f(x) = √2x + (-1+√2) and
f(x) = √2x + (-1-√2)

right?

I have yet to do the second one.

 

[Mark44]

Yes.