Some additional issues

Below we point out a few more characteristics in some Mathematica commands that you will use during the year. Execute each command, notice its output, and read the explanation that follows.

Here we have defined y as we had earlier. Now consider

Mathematica considers xy as a new variable, since we "forgot" to put a space between x and y.

Now execute:

x y

Mathematica remembers the definition of y and substitutes it.

Expand[%]

In this Expand command, the percent sign (%) stands for "the last result". Therefore, the product x y is expanded here to get a fourth degree polynomial.

Factor[%]

Likewise, this command factors the last result. Finally, a bit more subtle command:

y /. x -> 2

The effect of this command is to evaluate y at x = 2.

It illustrates a very important feature of Mathematica. You should read this command as "evaluate y subject to the rule: x is set to 2". The operator /. is made up by a slash / followed by a period (.) and the arrow is a "hyphen" followed by a "greater than" sign (>). The spaces after the y, before the x, and before the 2 are NOT necessary here, but they help in the readability of the command.

Defining a function in Mathematica

As you probably noticed, the last example is an awkward way to evaluate y at x = some value. An easier way to do such evaluations is to use the "function" notation. The following is an example of a function definition for f :

f[x_] := x^2 + 1

There are several items to note in the above command that are peculiar to Mathematica and that you must include in this definition to make f a legitimate function:

square brackets must be used to enclose the argument of f,
an underscore (shift on the hyphen) follows the argument inside the brackets, and
the assignment operator := replaces the usual equal sign.

Examples using a defined function

Some advantages of the function notation are shown in the examples below. These examples give you an idea of the power you have in Mathematica when f is defined as an actual function. The original argument, x, can be replaced by any numerical value or symbol. EXECUTE the following commands, one at a time, starting with the definition of f .

f[x_] := x^2 + 1

f[x]

f[3]

f[-3]

f[-12]

f[p]

f[cat]

f[t^3+1]

Expand[%]

As you can see, the original argument, x, can be replaced by ANYTHING! For example, with the argument t^3 + 1 we obtained a sixth degree polynomial (the "composition" of the functions x^2 + 1 and t^3 + 1).

Working with a Rational Function

You can form a so-called "rational function" by taking the quotient of y and f[x]. Execute the following command to define the ratio of y and f[x], and then execute the second command to plot the function.

r = y/f[x]

Plot[r, {x, -20, 20}, PlotRange -> All]

You can see here that r has the same general shape, for "small" values of x, as did y. (Why is this?) Also notice that for "large" values of x the graph of r looks linear. (Why is this?) To explore the second question graphically we plot functions r and x-1 on the same graph as follows. Execute the command:

Plot[{r, x-1}, {x, -40, 40}, PlotRange -> All]

In this tutorial, you have seen demonstrated two phenomena that are quite common:

Most functions look linear if you "zoom in" very close; that is, if you look at the function over a very small (domain) interval.
Some (but not all) functions look linear if you look at them over a large (domain) interval, or "zoom out".

The concepts of "very small" and "very large" play central roles in calculus, as you will see this semester.