1.4.1 Doing Versus Being

So far, we have stressed the process interpretation of function: A function is a rule for doing something to each number in a certain set of numbers (the domain) to get some particular result:

• Put in an `x`, get out a corresponding `y`.

• For each time `t`, find a distance `s`.

• Process each allowable input to produce the corresponding output.

• For each entry in column 1, find the corresponding entry in column 2.

• For each `x`-coordinate, go up to the graph, then over to the `y`-axis to find the corresponding height.

Each such function rule contains one or more active verbs that collectively specify an action to be carried out on each number in the domain.

Now we turn our attention to functions as objects. You should think of the grammarian's meaning of the word “object”: the noun that receives the action of the verb. Functions will often be the objects to which we do things. That is, the operations of calculus will be ones we carry out on functions to produce other functions. Thus, it is important to understand a function as a single object that can itself be the input to some process.

The object interpretation of function is important because the answers to our questions, the solutions to our problems, will often turn out to be functions or collections of functions, not just numbers or particular values of variables.

Whatever a function is as an object, it is obviously a more complicated object than a single number or a single variable or even a small collection of these things — the sorts of things you are used to seeing as answers. On the other hand it is not a more complicated object than a graph. Indeed, many graphs are just pictorial representations of the very objects we have called functions. Thus, if you start from your mental image of graph as an object, you won't be too far away from having a mental image of function as an object. The hard part will be to connect the geometric image with algebraic properties of functions.

Keep in mind the characteristic that separates functions from relations in general: the fact that each first coordinate is paired with exactly one second coordinate. We can use that characteristic property to decide whether a particular graph is or is not a representation of a function.

Activity 1

1. For each of the graphs in Figure 1, decide whether the graph is or is not the graph of a function. If it is not, state explicitly what keeps it from being the graph of a function.

2. In one complete sentence, state in your own words a simple geometric way to decide whether a graph is or is not the graph of a function.

Comment on Activity 1

In order to discuss algebraic properties of the objects called functions, we need some notation. It is easy to talk about processes in words, but it gets very awkward to do algebraic operations with words alone. Thus, we abbreviate words and phrases to single symbols. In a given discussion, each function is usually abbreviated to a single symbol, such as f (for function) or g (next letter after f) or $\phi$ (phi, the closest thing to f in the Greek alphabet). The independent (input) and dependent (output) variables for a given function are also abbreviated to single characters, such as t for time and s for position.

Using such a shorthand, we abbreviate the entire sentence:

The position of a falling body is a function of the time it has been falling.

to

The input to a function is inserted in parentheses following the function name, and the combined symbol is read “f of t.”

Contents for Chapter 1