The function $f(x)=a x+b$ is such that for all real $x$, $f(f(x))=x$. What is the geometric transformation (self-mapping) of $f(x)$ on the number line?
The original problem statement did not explicitly state that we are looking for a mapping of the number line onto itself. Therefore, we also accepted solutions that mapped the number line to another line in the plane. In the following solution, $f(x)$ represents the mapping of the number line onto itself.

According to the conditions for $f(x)$, we have $a(a x+b)+b=x$, which means $\left(a^{2}-1\right) x+b(a+1)=0$ holds for all real $x$. This is true if and only if

$$
a^{2}-1=0
$$

and

$$
b(a+1)=0
$$

From (1), we get $a=1$ or $a=-1$.

If $a=1$, then from (2) we get $b=0$. In this case, $f(x)=x$, which is the identity mapping.

If $a=-1$, then from (2) we get that $b$ can be any real number. In this case,

$$
f(x)=-x+b
$$

What kind of mapping of the number line onto itself is this? It is easy to see that the points with coordinates $x$ and $-x+b$ are symmetric about the point $\frac{b}{2}$, since $\frac{x+(-x+b)}{2}=\frac{b}{2}$.

Therefore, the mapping (3) is a central reflection of the line about the point $\frac{b}{2}$.

Thus, we can answer the question of the problem by saying that $f(x)$ is either the identity or the reflection about the point with coordinate $\frac{b}{2}$.

Remark. In our problem, we have a mapping with the following property: If the image of a point $P$ on the line is $Q$, then the image of the point $Q$ is $P$. This means, in other words, that the mapping is equal to its inverse. A non-identity mapping of the line onto itself with this property is called an involution. An example of an involution of the line is the mapping $f(x)=\frac{a}{x}$.