A translation of the reals is a function f R rightarrow R su

A translation of the reals is a function f: R rightarrow 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. What is the composition h compositefunction g compositefunction f of three functions: a reflection f, a translation g, and a reflection h? If you cannot work with arbitrary functions at first, consider some concrete examples.

Solution

Consider a simple example ---

F(x) = 5x

Now, reflection is----

f(x) = (x+5x)/2 = 3x

Now, according to given definition of translation ------

g(f(x)) = f(x)+k = 3x+k here k is a constant.

and now again an another reflection ------

h[g(f(x))] = hogof(x) = (3x+k+x)/2 = 2x + k/2