How can I tell if something is a function by just the X and Y values?
A function must assign each X value to exactly one, and only one, Y value. If there is an X value that appears multiple times in your list of X values, and the corresponding Y values are not the same, then it is not a function. The reason for this is because a function must be well-defined. You want to know exactly what the result will be if you put in an X value. So if there are multiple options for Y when you input a certain X value, then you won't know which one to use. It is fine for one Y value to have multiple X values associated with it, though, so be careful!
What is a derivative?
A derivative is the rate of change of a variable. Let's consider a variable that represents your distance from home as you are driving your car to school. As you travel closer to school, your distance from home increases with time. Chances are, though, that you do not drive at the same speed the whole way. You might go slowly through the neighborhood, faster on main roads, and then slow down again closer to the school. The speed that you are traveling is the derivative of the distance from home. Your speed is a measure of how your distance is changing--when you drive at a faster speed, your distance from home is increasing at a faster rate. Thus, speed is the rate of change of the distance, and that's the derivative!
What are Type I and Type II error?
Type I error occurs when we reject a true null hypothesis. You can think of this case as someone who is innocent has been found guilty. Type II error occurs when we fail to reject a false null hypothesis. You can think of this case as someone who is guilty get away with the crime.