Many students seem puzzled by function notation well after it has been introduced, and ask “why can’t we just write instead of ?”. To motivate the use of function notation and improve understanding, I advocate using multi-variable functions instead of single-variable functions in introducing this notation. My introduction usually proceeds something like this:
Suppose you are talking to a friend over the telephone, and they have a piece of paper in front of them on which they have drawn a circle using a ruler and a compass. You need the clearest, most concise instructions possible so that you can exactly duplicate the appearance of the paper your friend has… your circle must be exactly the same size as theirs, and in the exact same location on the piece of paper. What information do you need in order to be able to do this?
For the size of the circle, either the radius or the diameter will work nicely (most students come up with this answer quickly). If you are using a compass to draw the circle, the radius will probably be more convenient. However, the radius alone does not tell you where to locate the circle on the paper. What aspect of a circle will do the best job of telling you where it is located (this question is often challenging for students to answer initially)?
The center of a circle is the most useful attribute for identifying its location. The location of a circle’s center on a piece of paper can be described using a coordinate pair (assuming the bottom left corner of the paper is the origin, and the coordinates are given in centimeters).
So, you will only need three numbers to duplicate the circle. Suppose your friend tells you three numbers: “3, 2, and 1”. Can you duplicate their circle with that information? Why, or why not?
There is a small problem… your friend has not told you what each of the numbers represents. This is the first big reason for using “function notation”. If someone asks you to draw
Circle(3, 2, 1)
without giving you any other information, you don’t have enough information to be certain you are drawing the correct circle. Which of the three numbers is the radius? Or is it a diameter? Which is the x-coordinate of the center? Etc.
What is missing is the “function definition”. The function definition tells you what the function does, how many parameters (or “arguments”) the function requires as inputs, what those parameters are, and in what order they occur. In this example, the following information is probably what you wanted to know:
Circle(x, y, R) draws a circle with center at (x, y) and radius R
Now you have enough information to interpret “Circle(3, 2, 1)” correctly. The definition tells you the name of the function (for when it is referred to later), how many parameters have to be supplied (three in this case), the order they appear in the parameter list that must be used when the function is invoked (x-coordinate, then y-coordinate, then Radius), and what the function will do with that information.
Function Notation in Math Class
Turning to how function notation will typically be used in math class, you will usually see function definitions that look like this:
Or, in English: there is a function called “f” (instead of “Circle” as used in my example above) which requires only one parameter, which is labelled “x”. You will be asked to evaluate the function “f” for a particular value of “x” using this notation:
This can be read as “what is the value of “f” when its parameter is 4?”. To answer such a question, refer to the right hand side of the function definition above, which tells you to multiply the first (and only) parameter by three, then add two. So, in this example
Furthermore (thanks to Christopher for pointing this out) when you begin to talk about more than one function at a time, as in “compare these two graphs” or “which function reaches 100 first?”, it becomes very useful to have a name for each graph or function. If we had defined them both using “y =”, we would be stuck trying to distinguish between them using potentially ambiguous phrases like “the first equation” or “the curve with the highest y-intercept”.
If equations are defined using function notation, for example
the notation provides names for us to use when referring to them. We can then use “f” and “g” to label each on the graph, refer to them in conversation, and/or invoke them on specific values. This eliminates much confusion when comparing multiple functions, talking about functions of functions such as f(g(x)), or referring to functions defined previously.
Composition of Functions
There is one additional notation you should know about: the symbol used to indicate “composition of functions”, or “taking a function of a function”. The following two expressions are equivalent:
So, when you see the open circle between two function names, evaluate the one to the right of the circle, then plug its result in for the variable in the function on the left of the circle. Or, if you prefer the notation on the left of the equal sign above, you are welcome to convert the expression to that notation.
For example, using the definitions of “f” and “g” from above:
Alternatives to Function Notation
What would we do if function notation did not exist? We would have to invent one or more alternative notations. Yet, we already use such notations all the time.
A single (unary) minus sign tells us to negate a quantity. If we were to define it using function notation, it might look like:
Neg(x) = the value which, when added to x, will equal zero
Instead of using the plus sign to represent addition, we could define the Add function:
Add(x,y) = the arithmetic sum of x and y
Instead of using exponent notation, we could define the Square function as:
Square(x) = the result of multiplying x by itself
However, the notation we use for the most frequently used operations is much more convenient than function notation. Consider for a second how polynomial functions would look in the absence of exponential and arithmetic notation:
So, function notation is wonderful for complex or infrequently used functions. However, it can get in the way for simple operations or ones that are used frequently.
Function Notation in other subjects
There is one more wrinkle that is useful to know about function notation. You may occasionally see function definitions such as:
If you read through the above carefully, you will notice that there are two variables in the definition that are NOT shown in the parameter list. Shock and dismay! What to do? This situation usually occurs when there are constants involved (“a” and “b” in the above example) which people prefer to identify by name instead of by their value. For example, the constant (pi) is almost always referred to by its Greek letter instead of its decimal approximation, because this allows the person using the constant to fill in as many decimal places as they need in their calculation, and also because this gives someone reading the function definition a better idea of what this constant represents (where it came from).
Any variables in a function definition that are NOT included in the parameter list should be constants that you already know from the context in which the function is being used (Physics and Economics are two areas where this happens quite often).
Why is it f(x) so often?
Why don’t math teachers give their functions more interesting names, instead of “f”, “g”, or perhaps even the exotic “h”? Usually because they would like to be able to write the function name quickly and clearly, and because the function is an abstract one… it does not actually correspond to a real-world situation. For example, a function that describes the height of an object above the ground (a projectile motion problem) might be defined as follows by a math teacher who is not discussing any of the physics behind it:
and as follows by a physics teacher
where “g” is a constant used to represent the acceleration due to gravity, “” is the initial velocity, and “” is the initial height. The initial velocity and height values might be included as variables in the definition in order to remind all students of the role those constants play in the definition, even though their values are given immediately below. A teacher who is going to have to write the name of the function on the board many times over the next few minutes is much more likely to pick a single letter function name (like “H” above, which can be written very quickly and clearly) than a word (like “Height”, as might be used in a computer programming language).
But why is f(x) used so often? I think mostly because “f” is the first letter of the word “function”, and when a teacher is making up a random function to illustrate a point, they do not wish to spend too much time being creative with the name of the function, thus “f” leaps onto the board. So, the next time you see
you may still roll your eyes at the use of “f(x)” instead of “y”, but you will hopefully now understand why this notation exists, what sorts of situations require it to be used, and how to interpret it. From a teacher’s perspective, I would much rather write the above definition then ask you to evaluate
than have to write out
…Please calculate the value of “y” when “x” is 5…
Notation is a shorthand, a concise way of representing relationships or operations. Mathematics has many notations, including the arithmetic operators (plus, minus, times, divided by), powers, roots, logarithms, sigma notation, pi notation, and others. Function notation is yet another notation, one which you will see used many times in quantitative subjects like biology, chemistry, physics, economics, probability, statistics, and mathematics.