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One definition of function puts the term in the category of role playing -- activities assigned to, or required of, or expected of a person -- the person's expected rules of behavior, a model. There is a sense of duty, of what one is supposed to do, like the role of a parent.

You can also think of a function as a rule that associates one thing with another, or that assigns one value to another according to the value of the first. For instance, activities might be assigned to moments of time, like in the army, at work, or in personal scheduling.

Similar relationships define functions in math, except assignments are numerical. They are numerical functions. Variables are related to other variables in special ways to form the functions, and the relationship may be one-to-one or many-to-one. When the relationship is one-to-one, we have a single-valued function of one variable. In this case a particular variable (now called the independent variable) is so related to another variable (called the dependent variable) that for each value assigned to the first, there is one assigned to the second.

For example, the money you collect selling birthday cards depends on how many you sell. Double the number sold, double your money. Money is the dependent variable, and the number of cards sold is the independent variable.

On the other hand, when the relationship is many-to-one, we have a single-valued function of multiple variables. You then get specific values for the function (the dependent variable) only when you assign specific values to each of the many independent variables.

For example, the money you acquire may depend not only on the number of cards you sell, but also on the number of other goods sold, as well as the bills and taxes you pay. Money, here, is still the dependent variable, but now you have several independent variables to contend with.

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Suppose the independent variable x has the values:

Now, let's associate each of these numbers with others from set y and connect the number 2 with the number 1. Then connect the number 7 with 2. Then 4 with 3. And so on, as the rows of the following table:

In this format, each row expresses a specific pairing of numerical data and is the equivalent to a sentence that provides information about the function. The information provided by the first row (after the title row) is:

From row 2 you get:

And so on.

The general form can obviously be summed up in the expression:

The general form isn't very revealing, but one thing you can say is that each x is connected with, or goes to, only one value of y. The number 1 in x, for instance, maps to number 2 in y, and only to the number 2. Similarly, number 2 in x goes only to number 7 in y. And 3 in x only goes to 4. And so on. This is characteristic of single-valued functions. Given the value of the first variable, the value of the second variable is determined uniquely.

We can also represent the function as a graph. This is a different interpretation -- a different way of reading information. The graphical structure in the case of a function of one variable is called the two-coordinate graphical system. In this case values for the independent variable x are represented along a horizontal coordinate, and corresponding values of y are represented along a vertical coordinate.

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Functions of one form or another have characteristics that show up as patterns on graphs. Linear functions, for example, are straight lines. In particular, linear functions of one variable show up as straight lines on a plane surface.

A linear function of one variable takes values that are determined strictly by one variable. So the function:

is a function only of x. It is a linear function if the number 1 is the largest exponent of x in the expression.

So, for example,

is linear.

We can list some of the values of x and y in a horizontal table, as follows:

In words, reading down the columns, if x is 0, then y is 4. And if x is 2, then y is 10. And when x = -1, then y = +1. And so on. I'll leave it to you to plot the graph and explain to yourself that

is the inverse function of y and that it, too, is linear.

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 The quadratic equation has the general form:

The left side of this equation is the quadratic function. It is called quadratic, because there is an exponent equal to 2 and this is the highest exponent. To see its characteristics, consider an example. For simplicity sake, set a = 1, let b = 2, and take c to be 1, as well. We then have:

Sample values for x give us the following table:

There is no inverse to our quadratic function, y. You can spot this immediately by the fact that, for instance, y = 4 connects, not to only one number in x, but two, namely 1 and -1.

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To talk about deterministic functions, now, is, first, to talk about functions of time (t), namely:

The reference is to things happening in time. Plug in a value for the time and you get a value for the function. If t = a, then y_a = f(a). If a ball is rolling down a hill, it will have a certain speed, y_a, after a certain time, a. The function provides a model of the relationship and thus determines the rule.

The time function is deterministic, or not, depending on whether or not it is mechanical or causal in nature. To say that a system is deterministic is to say you can specify precisely what the value of the function will be for any particular time in the future you specify. To that extent, the function expresses something predictable. You can say precisely where the ball will be after a certain time. It won't suddenly have jumped out of line -- unless of course it was struck by something and caused to veer off course. Given that cause, though, even the off-line motion is predictable.

So there is a sense of mechanism. Or cause. Or determination. When things are caused to happen, they are said to be deterministic and can be predictable. If a certain amount of time has elapsed since a ball is released, the ball will have rolled a specific distance downhill. The cause is a sufficient condition, A, for the occurrence:

If A happens, B is bound to follow. There is no chance factor, or randomness, to alter the causal status. The deterministic function describes that causal phenomenon. 

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