We have learned that if you let $P$ be the Euclidean plane with distance $d$, a function $F: P \to P$ is an isometry if, for all points $X$ and $Y$ of $P$, $d(F(X),F(Y)) = d(X,Y)$. Also the following types of transformations are isometries: translation, rotation, reflection, glide reflection. The identity transformation is the function $F$ defined by $F(X) = X$ for all $X$. In other words, for all points $X$ the transformed point $X'$ equals $X$. A translation with translation vector $0$ is the identity. A rotation with rotation angle $0$ is the identity.
My question is that if a translation of the reals is a function $f:\mathbb R\to\mathbb R$ such that there is a constant $b$ so that
$f(x)=x+b$
for all reals $x$ . The reflection of the real line in a point $u$ is the function $f(x)$ such that $u$ is the midpoint of $x$ and $f(x)$ for all $x$.
How do you find the composition $h\circ g\circ f$ of three functions: a translation $f$ , a reflection $g$ , and a translation $h$ ?
We write the translations $h,f$ as $h(x)=x+a$ and $f(x)=x+b$. We can also write $g$ as $g(x)=2u-x$, where $u$ is the unique stationary point of $g$. The composition is given by $$(h\circ g\circ f)(x)=h(g(f(x)))=h(g(x+b))=h(2u-(x+b))=(2u-(x+b))+a$$ This is $$2u-b+a-x=2\left(u-\frac{b-a}{2}\right)-x$$ which is a reflection in the point $u-\frac{b-a}{2}$