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text: '# Even and odd functions\n\nRecall that a function $f : \\mathbb{R} \\to \\mathbb{R}$ is [even_function even] if $f(-x) = f(x)$, and [odd_function odd] if $f(-x) = - f(x)$. A typical example of an even function is $f(x) = x^2$ or $f(x) = \\cos x$, while a typical example of an odd function is $f(x) = x^3$ or $f(x) = \\sin x$. \n\nWe can think about evenness and oddness in terms of [3g8 group theory] as follows. There is a group called the [cyclic_group cyclic group] $C_2$ of [3gg order] $2$ acting on the set of functions $\\mathbb{R} \\to \\mathbb{R}$: in other words, each element of the group describes a function of [3sz type]\n\n$$ (\\mathbb{R} \\to \\mathbb{R}) \\to (\\mathbb{R} \\to \\mathbb{R}) $$\n\nmeaning that it takes as input a function $\\mathbb{R} \\to \\mathbb{R}$ and returns as output another function $\\mathbb{R} \\to \\mathbb{R}$.\n\n$C_2$ has two elements which we'll call $1$ and $-1$. $1$ is the identity element: it acts on functions by sending a function $f(x)$ to the same function $f(x)$ again. $-1$ sends a function $f(x)$ to the function $f(-x)$, which visually corresponds to reflecting the graph of $f(x)$ through the y-axis. The group multiplication is what the names of the group elements suggests, and in particular $(-1) \\times (-1) = 1$, which corresponds to the fact that $f(-(-x)) = f(x)$. \n\nAny time a group $G$ [3t9 acts] on a set $X$, it's interesting to ask what elements are [invariant_under_a_group_action invariant] under that group action. Here the invariants of functions under the action of $C_2$ above are the even functions, and they form a [subspace] of the [vector_space vector space] of all functions. \n\nIt turns out that every function is uniquely the sum of an even and an odd function, as follows:\n\n$$f(x) = \\underbrace{\\frac{f(x) + f(-x)}{2}}_{\\text{even}} + \\underbrace{\\frac{f(x) - f(-x)}{2}}_{\\text{odd}}.$$\n\nThis is a special case of various more general facts in [3tn representation theory], and in particular can be thought of as the simplest case of the [discrete_Fourier_transform discrete Fourier transform], which in turn is a [mathematical_toy_model toy model] of the theory of [Fourier_series Fourier series] and the [Fourier_transform Fourier transform]. \n\nIt's also interesting to observe that the cyclic group $C_2$ shows up in lots of other places in mathematics as well. For example, it is also the group describing how even and odd numbers add<sup>1</sup> (where even corresponds to $1$ and odd corresponds to $-1$); this is the simplest case of [modular_arithmetic modular arithmetic]. \n\n<sup>1</sup><sub>That is: an even plus an even make an even, an odd plus an odd make an even, and an even plus an odd make an odd.</sub>',
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