Function Notation
or
"OK, so f(x) doesn't mean f times x, what does it mean?"

The Notation Game
We use f(2) to stand for the output that goes with 2 in the f function.
The letters f, g, and h are our favorite letters for functions.  (Other letters may be used, too.)  A function is a rule in which each input has exactly one output.  To determine an output all you need is the function, and an input.  Say the function is f and input is 2.  In mathematics, 
For example, f(2) = 7, then the input is 2 and the output is 7.
Keep in mind that outputs are often y-values, so we may use f(2) to stand for the y-value that  we get when x is 2, in the f function.  Furthermore, since inputs and outputs are often put into ordered pairs, we may use f(2) to stand for the second coordinate that goes with 2, in the f function.  For example, f(2) = 7, then (2,7) is an ordered pair in the f function.
f(x) is the output that goes with x in the f function.  Note that here, x is a varying quantity.  You may think of it as placeholder.  Usually, x can take on any of a number of values.
For example, if  f(x) = 2x+1, then the input is x and the output is 2x+1.  If the input is 6 the output is 13.   f(6) = 13.

So, isn't f(x) just y ?
f(8) is the y-value (output) but it also tells which y-value, namely the y-value that goes with 8 in the f function.  Often times y does not suffice because there may be lots of x-values and lots of functions.  (For example:  f(3), g(3), t(3), f(7), t(7), s(7.25).)  In some cases, you will find that there are also advantages to using (just) y instead of function notation ("f(x)"). 

Function Notation in a Different Disguise
The same function notation idea is used in many other settings, but since f, g, and h are not used, the similarity if often missed.  Three examples:
C log(x), sin(x), abs(x).  With frequently used functions (like logarithm, sine, and absolute value) a shortened form of the name of the function is used instead of f.  
C On an electronic (computer) spreadsheet @ is used to indicate functions.  For example, @AVG(2,5,9) means to output the average of 2, 5, and 9.  Here, three inputs were needed.
C Probability.  Say a coin is flipped.  The function is the probability function.  P(heads) = 2 means the input is heads, and the output is 2.  Capital P is used instead of f.

The idea of a function is central for understanding mathematics, science, and applications of mathematics.  Understanding the notation is a key for the understanding of functions.