Problem 5

Let

G

be a set of non-constant functions

f:\mathbb R\to\mathbb R

of the form

f(x)=ax+b

, with real

a,b

and

a\ne0

. Suppose: if

f,g\in G

then

g\circ f\in G

; each inverse

f^{-1}

belongs to

G

; and every

f\in G

has a fixed point. Prove that there is

k\in\mathbb R

fixed by every

f\in G

Step 1 of 4: Fixed points and inverses

In plain words

An affine map with slope one can have a fixed point only when it is the identity.

f(x)=ax+b, f^{-1}(x)=x/a-b/a

Detailed analysis

If $a=1$ , having a fixed point forces $b=0$ , so $f$ is the identity. If $a\ne1$ , its unique fixed point is $b/(1-a)$ .