## Friday, January 31, 2003

### Experience Mathematics # 26 -- Symmetries

There are two kinds of symmetries in a function. A function may be symmetric across the $y$-axis, or symmetric across the origin. (If a curve is symmetric across the $x$-axis, it is not a function. Can you tell why?)

For example, the function $f(x)= x^2$ is an example of a function that is symmetric across the $y$-axis.

This symmetry is obvious from the graph. An algebraic way to see that the function $f(x)= x^2$ is symmetric across the $y$-axis, is to replace $x$ by $–x$ in the formula, and note that:
$f(–x) =f(x)$ (since $(–x)^2=x^2$).
For example, the $y$-coordinate corresponding to the point $–2$ is the same as that corresponding to $2$.
The function $f(x)= x^3$ is an example of a function that is symmetric across the origin.

Each point (for example the point $(2, 8)$) maps to a symmetric point (the point $(-2, -8)$) in the graph. An algebraic way to notice that this function is symmetric across the origin is to note that
$f(–x) =–f(x)$ (because $(–x)^3= –x^3$).

Functions symmetric across the $y$-axis are called even functions, and functions symmetric across the origin are called odd functions.

What is remarkable is that any function defined on the set of real numbers can be written as a sum of an odd and an even function. Can you figure out a way to write the exponential function $f(x)=e^x$ as the sum of an even and an odd function? The curve formed by a hanging clothesline  appears in the answer to this question.

## 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.

## Friday, January 10, 2003

### Experience Mathematics # 24 -- The Calculus

Happy New Year. The earth has finished another revolution around the sun, taking a little more than 365 days to do so. Meanwhile, the moon continues to rotate around the earth, the planets around the sun, and the same forces that make these things move in an elliptical path ensure that a ball thrown up in the air always falls down, or that a ball thrown in the air (towards a friend) takes a parabolic path.

Over this and the next few columns, I will discuss these natural motivations that are behind the notions that you encounter as you study the Calculus.

The first concept is that of a function. Mathematicians were already familiar with curves from Euclidean and coordinate geometry by 1600 or so A.D. It was natural to begin modelling various physical phenomena with functions. For example, $y=1-x^2$ models the parabola. For each value of the input $x$, we get a unique output $y$. If you plot the curve in the coordinate plane, you obtain a parabola.

It was natural to do two things. To figure out laws that can explain why a ball thrown in the air follows a path traced by such a curve. This led to the laws of Gravitation. And the other thing is to use these laws to predict the answers to common questions that arise. For example: How high will the ball go? How far will the ball go? Given the curve, when does the curve go up (increase)? And when does it come down (decrease)? We will consider such questions and relate them to what you encounter in Calculus.

Curves such as the circle ($x^2+y^2=1$) are not functions since there is not one output $y$ for each input $x$. For example, for $x=0, y$ can be $1$ or $–1$. So, every curve does not give rise to a function.