Function is either Type 1) or 2) relationship. e.g.
a) y = 5x + 2 is a function because one value of x yields one value of y. Thus, this function is a one-to-one relationship. Point A is a point on the line y = 5x + 2. No matter how you move the position of point A, you will see that one value of x gives to one value of y.
b) y = x2 - 1 is a function because two values of x yields to one value of y for all values of x except at y = -1, which is produced by one value of x only, i.e. x = 0. Therefore, this function is a many-to-one relationship. Drag point A and you will find that point A1 has the same y-coordinate with point A but different x-coordinate.
c) y2 = x is NOT a function because one value of x yields two values of y. For example, when x = 1, we get y = 1, -1. Try to move point A. You can see that there is a corresponding point B (which has the same x-coordinate) has different value of y-coordinate from point A. Thus, this equation is a one-to-many relationship.
d) x2 + y2 = 25 is NOT a function because this is a many-to-many relationship. For example, when x = 3, we get y = 4, -4. On the other hand, when y = 4, we get x = 3, -3. Try to move point A and look at the changes of position of point B, C and D.
There are many types of functions. I listed out some of them with graphs provided. You may click on any topic under Function at the left side bar to have further understanding on Function and also the shifting/scaling/reflecting of the function.
Betty, Created with GeoGebra