The shape of the beast

The other day, right at the end of a tutoring session, a student sprang an awkward exam question on me. "Sketch the curve y = xˣ ", it said, or something like that. Surely that can't be hard, I thought. After all, y = x² is GCSE stuff, and y = eˣ is encountered early in the A-level course. So let's try it. It turns out to be surprisingly beastly, but we will tame it as much as we can. Fasten your seat belt.

For x ≥ 1 it's not that hard. The curve rises steeply: at x=1 it's 1, at x=2 it's 4, at x=3 it's 27, at x=4 it's 256, and then it takes off into the thousands and millions - faster than exponentially, because the base (the big x) is increasing as well as the exponent (the little x).

But what about fractional negative x values like -1/2? We want (-1/2)⁻¹/², which is √ (-2), which is pretty hard to plot on a normal x-y graph. On the other hand, x = -1/3 is not so bad, because negative numbers have well-behaved negative cube roots. Here's what Desmos makes of it. It only manages to figure out a few values for negative x; if we were to join the dots, it appears we would get the same curve reflected in the y axis, but Desmos is not joining the dots, and if you look closely, you will see that the x values it chooses for the higher and lower dots do not coincide. What the heck?

To make sense of this, observe that since x = (-1)(-x), xˣ is (-1)ˣ (-x)ˣ = (-1)ˣ / (-x)⁻ˣ . Since x is negative, the denominator after the slash is the same as xˣ for positive x - but we're using its reciprocal, so whereas the curve to the right of the y axis goes down and then way up, on the left we're going up and then way down towards zero. So we just need to work out what (-1)ˣ is for arbitrary x. Here's how we do it.

We know (if we have diligently studied the complex numbers part of the Further Maths A-level; apologies if you're just doing the one A-level) that -1 = cos((2n+1)π) + i sin((2n+1)π) for any integer n, so (-1)ˣ = cos((2n+1)πx) + i sin((2n+1)πx). We'd really like that to have zero imaginary part for it to appear on our plot. This means the expression inside the "sin" has to be an integer multiple of π; let's call it mπ. Thus (2n+1)πx = mπ, or x = m/(2n+1). If m is even, mπ will be a multiple of 2π and so cos(mπ) will evaluate to 1; if m is odd, it will evaluate to -1.

So it appears we will get a value we can use if x is a rational number which, when common factors are eliminated, has an odd-valued denominator. If the numerator is even, we get (-1)ˣ = 1; if it's odd, we get (-1)ˣ = -1. If x is a rational number with an even denominator, like 1/4, or an irrational number like √2, e or π, we're out of luck. But we can add points to our graph for as many rational x values (with odd-valued denominators) as we like. Desmos won't do this for us, so here is my shaky hand-drawn attempt:

Filling in the spaces between the dots is allowed because the rational numbers - even just the ones with odd denominators - are "dense" in the real numbers: if you give me any real number, I can give you a rational number with a odd denominator that's as close to it as you like (as long as you specify beforehand how close it has to be). So I'd argue drawing continuous curves is fine as long as my pen has a positive width - though you might equally argue that it's not fine at all, because y = xˣ is a function with one y value for each x value, so drawing two separate curves is not OK at all and I should instead draw lines going up and down between positive and negative values infinitely many times.

If you got this far, congratulations; you have tamed the beast, or at least you have watched it being partially tamed without running away. If you are working towards a maths or further maths A level, you might want to ask about some tutoring; with any luck we won't discuss anything quite this beastly, because although, as I said at the start, this article is loosely based on a practice exam question, the examiners were much nicer than me and didn't encourage anyone to wander off to the left of the y axis where the real horrors lurk.