Numbers & Functions · 7 Functions as Objects

7.2 Symmetry: even and odd, properly this time

We’ve brushed against even and odd twice now. Now we can say what they really mean — they’re statements about the flip. A function is even if flipping the input changes nothing: f(−x) = f(x). Its graph is a mirror image across the vertical axis. The squaring machine is even; so is cosine. A function is odd if flipping the input flips the output: f(−x) = −f(x). Spin it 180◦about the origin and it lands on itself. The cubing machine is odd; so is sine. Why give these names? Because symmetry is one of the most powerful laborsaving ideas in mathematics. If you know a function is even, you only have to understand half of it — the other half is a guaranteed mirror image, free. (And it’s a quiet first taste of the symmetry that Galois built group theory from, back in Chapter 4’s deep water.)

x f(−x) = f(x) even: mirror across the vertical axis x f(−x) = −f(x) odd: 180 ◦ spin about the origin Figure 7.3. Even versus odd symmetry. Even functions mirror across the vertical axis; odd functions map onto themselves under a 180◦rotation about the origin.

7.3 Building new machines from old

The transformations reshape one machine. The other way to make new functions is to combine machines. You can do arithmetic on whole functions, point by point: if f(x) = x2 and g(x) = x, then f + g is the machine x2 + x. (This is how polynomials were built. Division is the one to watch — f/g jams wherever g outputs zero.)

Composition — chaining machines. This is the big one, from Chapter 3: feed the output of one machine straight into another, f(g(x)). I’m bringing it back here because now you can see it as an operation that takes two machines and produces a third. And here’s the payoff: every transformation in the first half of this chapter is secretly a composition. Shifting f left by 3 is composing f with the tiny “add 3” machine. Stretching is composing with a “multiply” machine. That is precisely why composition is the heartbeat of modern mathematics. When an artificial intelligence processes something, it passes a number through an immense chain of simple machines, thousands deep. Every layer is a composition. You now hold the actual idea; the later books just stack it higher. Functions as Objects x ×2 2x f f(2x) . . . and a neural network is this, thousands deep Figure 7.4.

Every transformation is a composition. f(2x) is “multiply the input by 2, then apply f” — two machines in series. Stack this motif thousands deep and you have the architecture of a neural network.

All 8 chapters of Numbers & Functions · Foundations · All courses