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.

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