Section 1.3: Functional Notation and Terminology

In the last section, we introduced the function concept.  In this section, we will focus on the functional concept of evaluation.  We will then discuss terminology and notation associated with functions.  

Functional Evaluation

In the last section, we emphasized that a function is any clearly defined "rule" for coming up with an "output" for a given "input".  This process of taking an "input" to an "output" is called evaluation.  As we have already seen, such a "rule" can be specified in a variety of ways: 

  1. As a set of ordered pairs: for example, {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), ... }.
  2. As a table of values:
    3. x y
    -2
    -1
    0
    1
    2
    ·
    ·
    ·    		 4
    1
    0
    1
    4
    ·
    ·
    ·
  3. As an arrow diagram:

  4. As a 2-dimensional plot:

  5. As an equation: y = x2.
  6. As a "black-box":

For simpler rules, we may also be able to describe the rule in English:
  1. "Multiply the input by itself"
Each of these different ways of representing a function has its own strengths and weaknesses, and the precise mechanics of evaluation varies in each case.  Let's look at each case, showing how to evaluate the function given above on an input of 2:
  1. To "evaluate" with a list of pairs on an input of 2, we must search through the list for the pair with first element 2 and return the value of the second element in the pair: {(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4), ... }.  Using this view of the function is slow, and if the listing of pairs is incomplete, then it fails to work.  The only advantage of this point of view is that it highlights how the function is a mathematical relation.
  2. To "evaluate" using a table, we must search through the first column for 2 and return the corresponding value in the second column:

    This view has all the disadvantages of the ordered pair perspective.  It is mainly a convenient way to display numerical information.

  3. Using an arrow diagram is very similar to using a table: we must search through the first column for 2 and follow the arrow to the corresponding value in the second column:

    This view has all the disadvantages of the two previous perspectives.  However, it tends to take less space than a table and is helpful for quickly verifying whether or not the relation is a function.

  4. Using a graph is somewhat similar to using an arrow diagram.  We must search along the horizontal axis for the given input value, 2.  Then we move vertically, until we hit the graph (either up or down).  From that point on the graph, we move horizontally, until we hit the vertical axis (either left or right):

    The value at that point of the vertical axis is the "output" value, 4.  In some ways, this gives a more compete description of the function than any of the previous ones (although it is limited, since we can rarely draw the entire graph, which usually goes off to "infinity" on the ends).  However, we can only read off approximate values from a graph.

  5. If the rule is given by an algebraic formula, then evaluation is simply a matter of "plugging in" for the "input" variable and simplifying: y = x2 = (2)2 = 4.  This is one of the most convenient points of view, as long as the formula is not too complex.  It generally is the shortest to write, and yet is complete enough to work on any input value.
  6. If the black-box shows a formula, then this perspective is essentially the same as the previous one.

    Its main drawback is that you must perform the algebra somewhere else.  However, it more clearly displays the input and output values:

  7. A well-phrased, verbal description is quite helpful for some people.  In some ways, it is even easier than the algebraic view, since it should give a step-by-step description of how to take an input to an output.

If you think about it carefully, you will notice that functions are simply relations that can always be successfully (i.e., at least one output for each input) and consistently (i.e., at most one output for each input) evaluated.

Functional evaluation is one of the most crucial skills in this course, since it is the most basic thing you can do with a function.  Since we will alternate between these various points of view, so you should be comfortable evaluating a function using any one of our seven representations.

Practice evaluating functions by completing the following exercises.

Functional Terminology

To this point, we have been using the terms "input" and "output".  While these are intuitive terms, they are not standard mathematical terminology.  Instead, we use the terms independent variable and domain for "inputs", and dependent variable and range for outputs.  Specifically, consider the set of all possible "inputs"; in terms of each of our previous views of a function, this would be:

This set of possible inputs is called the domain of the function.  In our previous example, the domain is {all real numbers}.  We often associate this set of inputs with a variable; you can see that we used the variable x to label the first column of the table and in the the formula in our black-box picture.  This is known as the independent variable of the function, since it's value is free to vary among all the values in the domain of the function.  When we plot a function, the independent variable is associated with the horizontal axis.

In the same way, we often associate a variable with the set of outputs, such as how we used the variable y to label the second column of the table and on the the left side of the defining equation of the function, y = x2.  This is then called the dependent variable of the function, since it's value depends on (i.e., is determined by) the value of the input (i.e., the independent variable), according to the defining rule of the function.  When we plot a function, the dependent variable is associated with the vertical axis.  The dependent variable takes its value from the set of all possible "outputs".  In terms of our various views of a function, this would be: 

This set of outputs is called which is called the range of the function.  In our "square the input" example, the domain is {all positive real numbers}, since every positive number is the square of something, and the square of a real number is never negative.

It is possible to get a clear picture of the domain a function from its graph.  Simply trace along the graph with your pencil and highlight the points on the horizontal axis which are directly above or below your pencil.  You can picture the range in a similar manner, except you would highlight the points on the vertical directly left or right of your pencil.  The following picture illustrates this technique: 

Notice how the entire horizontal axis and only the positive part of the vertical axis end up being highlighted, showing the domain and range, respectively.

Make sure that you can identify the domain and range of a function, as described by a table, arrow diagram, or plot, by completing the following exercises.

Functional Notation

Just as we use letters as variables to stand for numbers, we also use letters to stand for functions.  This can easily lead to confusion.  There are three ways in which we avoid confusion:

By convention, we tend to use the letters f, g, h, and k to stand for functions, although we may occasionally use other letters.  Which is why it is important to pay attention to context.  Before we every refer to a function by a letter, we should clearly define the function and associate it with that letter.  We do this by consistently using functional notation.  For example, if we want to use the letter f to refer to the "square the input" function, we might say:

Let f be defined by the equation f(x) = x2.

Notice how, in functional notation, we put the independent variable directly to the right of the function name, and in parentheses.  This notation is supposed to suggest that we "feed" the variable into the function (from the right) to get an output; in other words, when see something to the right of a function name, we should think of evaluating the function.  Note: This notation was first used by Leonhard Euler (1707-1783) in 1734 (Florian Cajori, A History of Mathematical Notations, vol. 2, p. 268).

Given the defining equation above, evaluating the function f at x = 3, for example, then simply looks like substituting one equation into both sides of another:

When f is evaluated at x = 3, it takes the value f(3) = 32 = 9.

Notice that we have now introduced a new way to handle parentheses.  In Algebra, using parentheses in this way, such as y(x + 1), would indicate multiplication.  But when, from the context, we know that the letter to the left of the parentheses is the name of a function, then we must interpret this to mean functional evaluation!

This means that you must always pay attention to the context of any mathematical expression, so that you can understand how to correctly interpret the use of parentheses.

For example, if you know that s is a variable, then s(x + 1) would be simplified as multiplication.  In particular, if you were told that s = 3, then you could be sure that s is the name of a variable, and this would simplify as:

3(x + 1) = 3x + 3. 

However, if you were told that s is a function, then s(x + 1) would be simplified by evaluation.  For example, if we said:

Let s be defined by the equation s(t) = 2t - 5.

then it should be understood that s is the name for the function: "Multiply the input by 2 and subtract 5 from the result".  This time, the same expression as before would now simplify as:

s(x + 1) = 2(x + 1) - 5 = 2x + 2 - 5 = 2x - 3

Notice how, when substituting x + 1 in for t, we made sure to surround it by parentheses; this is a good habit, whenever substitution could lead to confusion (see Rules of Thumb).

Functional notation allows us to emphasize the fact that two variables are functionally related, even if we cannot specify the rule for the relationship!  For example, we know that there is a functional relationship, call it g, between the rate, r (in cycles/min.), at which you pedal your bike and the speed, s (in mi./hr.), at which you will travel (assuming that you don't change gears).  We can express this mathematically as s = g(r), even though we may not know the precise formula for the function, g.

This notation also makes clear which variable depends on the other.  In this example, r is the independent variable, since you can choose how fast you want to pedal.  Once you decide how fast to pedal, you are not free to determine how fast you will go, since s is then determined uniquely by the function, g ; in mathematical terms, s is the dependent variable.

Finally, although we may choose to assign different letters as names for different functions in different contexts, there are many functions to which we will always refer by the same name all the time, because these functions are so important and well-known.  In fact, these functions are so important that they have names that are longer than one letter in length, such as sin, exp, and abs.  The first such important function is also the simplest one.  It is called the "identity" function, its name is id, for short; it is the "do nothing" function:

The identity function is defined by the rule id(x) = x.  That is, it simply returns the input unchanged.

Notice that, as with function notation itself, such long function names could lead to confusion, unless we pay attention to the context.  For example, while one might be tempted to interpret an expression like id(5) as "evaluate the function d at 5, then multiply the result by the variable i", we must recognize from the context (such as the fact that we did not italicize the letters "i" or "d") that  "i" and "d" are not individual variable names, but part of a single function name.

Make sure that you understand these concepts by completing the following exercises.

Go to Functions and Algebra

Table of Contents Send questions or comments to jwicks@northpark.edu Glossary