You are right. The letters x and y should be inverted. What the sheet says is that the both x and y depend on the 'parameter' t. therefore you can write both x and y as functions of t. Hence the definition of x=f(t) and y=g(t).
For example, you can think of x and y as the coordinates of motion of a particle in two dimensions. As the particle moves in time, the coordinates of the point, that is (x,y), change. I think your prof just mistakenly inverted x and y.