Thursday, January 23, 2003

Experience Mathematics # 25 - Functions

A ball thrown in the air follows the path of a parabola. Parabolas are modelled by a function of the form $p(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are real numbers. This kind of function—a polynomial of degree 2—is called a Quadratic Function. While we will not formally define functions, it is helpful to get an intuitive idea of functions from several points of view.

One point of view is to think of functions as a rule. For example, consider the quadratic function:
$f(x)=1-x^2$. Every real number $a$ corresponds to a unique real number denoted by $f(a)$ obtained by replacing $x$ by $a$ in the above equation. For example,
$f(0)=1, f(1)=0, f(-2)=-3.$

This suggests that we can also think of a function as an input-output machine. For each input $a$ we have a unique output $f(a)$. The set of possible input values (in this case the set $R$ of real numbers) is called the domain of the function.

Imagine making a table of all the input-output values of the function. (There are an infinite number of elements in the domain, so you can only imagine making a table!) All these values can be plotted on the coordinate plane. The input values are the $x$-coordinates and the output values are the $y$-coordinates.

If we do this, we will get a graph of the function. We denote the graph by $y=f(x)$, (or $y= 1-x^2$).

This is the third way of thinking about a function: as a graph. The graph is shown below.

Note that this parabola is symmetric about the $y$-axis. It meets the $x$-axis when $x=1$ and when $x=-1.$ These are (graphically speaking) the solutions of the equation $1-x^2=0$. The function has a maximum when $x=0$, corresponding to the highest point a ball reaches, when it is thrown in the air.

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