Q: What is the difference between f, f(x), f (2), and 2f (x)?
A: Of course all of these deal with some aspect of functions. However, only the first expression, f, means "the function f ", i.e. a name for the rule that assigns an output to every input x. The second expression, f(x), is the notation for the output, i.e. the value of the function f at x. Usually, this value is given by a formula. The notation f (2), means "the value of the function f at the point x = 2". Finally, 2f (x), means "double the value of the function f (x) at every point x".

Q: Why is a circle not the graph of a function?
A: The graph of any function must be able to pass the "vertical line test". That is, at any place that you place a vertical line, it will hit the graph at one point at most. Now look at any circle. You can easily find a vertical line that "cuts" it in two - and in particular that touches the circle at 2 places. As long as there is one such line, the graph does not describe a function.

Q: If y = x² is  a function, is x = y² a function?
A: That depends. Strictly speaking, one must say "y is a function of x" in the first case. In the second, y is not a function of x (for example, the graph will not pass the vertical line test), but x is a function of y.

Q: How can I remember the order of operations in shifting/translating a graph?
A: These are very similar to the order of operations in algebraic expressions. For example, consider what you would do if you were to evaluate the expression 2(x + 3) - 1 at x = 4. You would first add 4  + 3 = 7, then multiply by 2, and only after that you would add -1. When you graph the function, the order is the same: first shift left by 3, then scale by a factor of 2, and last shift down by 1.

Q: The data for a parabola can be found in more than one way. What if I compute it in 2 different ways and come up with different answers?
A: This is actually why it is worthwhile to compute data in as many different ways as possible! All the different methods should yield the same results. If the results differ from each other, you must check your information and your computations, and find out which one is right. In the end they must agree with each other.