You could think of it as x is the GLUT output, f(x) is the subject's response, and g(f(x)) is the GLUT's interpretation of the subject's response. f maps from GLUT output to subject response, and g maps from subject response to GLUT output. f and g don't have fixed points, because they don't have the same domain and range. f∘g, however, maps from GLUT output to GLUT output, so it has the same domain and range. I was just calling it f, but this way it might be less confusing.