I have been asked
Does the set of all reflections in the plane form a group? Explain.
I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I have a shape that is more of an "n" shape rather than a "u" shape.
However in my notes I have found that when listing the elements of a symmetry group of the same parabola, a reflection in the y axis exists.
Would someone be able to help clarify where I am going wrong? It could be with understanding the concept of a group, or what it means to reflect a function, or indeed both. Thanks for any assistance available