The purple function is f(x), the blue function is g(x) and the red function is f(g(x)).
Can you define f(x) so that f(g(x)) = h(x), at least over a restricted domain?

A famous problem: given locations A and B and a line, to find the location C on the line which minimizes |AC|+|CB|. (This can be phrased as locating a pumping station on an oil pipeline to service two cities.)

The ellipse isotomic generated by point D is teh locus of the reflection of D in the tangents to the ellipse.
What shape is the isotomic when D lies on one of the foci of the ellipse?
Why?