Geometry of graphs: Monotonicity and symmetry

Let’s now discuss monotonicity properties of functions. The word monotonicity refers to increase or decrease. A function is called increasing if it satisfies this condition. Roughly speaking, if the argument is bigger, then the value of the function is bigger. It’s called strictly increasing if the inequality is strict. We can turn for an example to our affine functions that we discussed before. A function of the form mx plus b is increasing if and only if its slope, m, is a positive number. If m is positive, you have an increasing function. If m is negative, you have what is called a decreasing function. And if m is 0, you have a constant function.

Let’s examine now the graph of the function g that we saw before, g of x equals x squared. We give ourselves some coordinate axes. We plot a few points. And we trace the curve of the graph. And we come to something very famous, a curve whose name is in the dictionary. It’s called a parabola. Satellite dishes, incidentally, are of parabolic shape for a very focused reason that I won’t get into. Now this function g is an even function. That means it assigns the same value to minus x as it does to x. What that means for the graph is that the graph displays a certain symmetry. The graph is symmetric, we say, with respect to the y-axis.

The meaning of this? If we look at the right-hand branch of the graph, when we rotated around the y-axis, we come upon the left-hand branch. This is symmetry with respect to the given axis. So here’s our function, and there’s its graph. I ask you, is g an increasing function? Well, clearly not. g decreases when you’re on the left of 0, and it increases when you’re on the right. However, let us observe that if we were to restrict attention to the positive x’s, then it would be an increasing function. So we could restrict attention to R plus, as it’s called. Standard notation– R plus means the positive number’s in R, the interval 0 infinity. More generally, we have the following proposition.

For any positive integer n, the function phi, defined to be the n-th power of the argument, is a strictly increasing function on R plus. Here, of course, x to the n, or x power n, n-th power of x means x multiplied n times.

That proposition we’re now going to prove because we haven’t had the fun of doing a proof in a little while. Here’s the proof. It’s by induction. We’re going to first prove it for n equals 1. We want to prove that the function x maps to x is strictly increasing on R plus. That amounts to checking that x less than y implies x less than y. Well, that’s pretty evidently the case. So the case n equals 1 has been disposed of. Next, and finally, we have to prove that whenever x to the power n is strictly increasing on R plus, the same is true of the next power, n plus 1. Well, take two numbers, x less than y positive.

We must show this inequality. That inequality is certainly true if x is 0, so let’s assume x is strictly positive. Now we apply the induction hypothesis, the truth for n. So we have x n less than y n. If we take that strict inequality and multiply it by the strictly positive number x, we obtain this, which I’m going to save for later use. Similarly, if you take the same inequality but now multiply it by y to the n, you get this inequality. Now let’s combine the two inequalities we’ve obtained in a string of relationships. And you can see that we have the first, and we have the second.

And if you now look at just the extremities of these relations, you see that we have proved that x power n plus 1 is less than y power n plus 1, which is exactly what was required to prove. So that’s the end of the proof. They used to say QED at this point. Now they put this symbol, so perhaps we should say box or end of proof.