We’re going to spend some time looking at functions defined by powers. As is often the case, we begin with integers. Integer powers. For example, we’ve seen the function x squared in the past. We need some terminology about how things work with powers. If you have an expression like a to the power n, a is called the base, and n is called the exponent, or the power. Some facts about these concepts are the following. Let’s fix two integers m and n positive. Then we have all these rules for powers, which are actually easy to prove in the case of integers. I’ll let you look at them for a moment. Perhaps, they look familiar to you.
If you were to prove them, they would follow from the commutative law in arithmetic and the associative law that we saw some time ago. Now, these facts need to be second nature to you. If they’re not already, well, doing a million exercises will really help you remember them. Two specifications, a to the power 0 is always defined to be 1 provided that a is non-zero. And 0 to the power 0 is just not defined. There’s no way to define it so that things work out. What are the graphs of these power function’s going to look like? We already know two of them– x squared the parabola, x cubed that we saw not long ago.
We see that for n even we have a function that is increasing on r plus, whereas for n odd x cubed, we have a function, which is globally increasing. Also, the x squared was an even function. The x cubed was an odd function. Well, these things carry over to other powers. For example, x to the fourth is an even function, which is increasing but only on r plus. It looks a lot like the parabola, but you notice it’s much flatter near 0. It’s not a parabola. x to the fifth has roughly the same nature as x cubed, and so on.
If you zoom in a bit on the first quadrant, you see that the two functions, xm and xn agree at x equal 1. All of these functions do. They have the value 1 there. The x to the power m– the smaller of the two powers– well, it’s higher than the other when you’re between 0 and 1. But then it’s lower when x is greater than 1. How about negative integer powers? We can define those. Here’s how it’s done. We take an a different from 0 to avoid division by 0, and we define as usual a to the 0 equals 1.
And we define a to the power n where n is a negative integer by definition is equal to 1 over a to the power minus n. That’s already defined because then minus n is a positive integer. Now, why do we define a to the 0 equal 1? Well, here’s one explanation. We would like this law of exponents to still work for 0. And from what I’ve written down, you can see that this forces a to the power 0 to be the identity element for multiplication that is 1. Otherwise, this law would fail to hold.
Similarly, we have to define a negative exponent base this way because if we want this law to hold, then a to the minus n has no choice but to be the reciprocal of the other value. For example, what is pi to the power minus 2? It turns out to be 1 over pi squared. What is pi to the power– minus pi, rather, to the power minus 3? Answer, it turns out to be minus 1 over pi cubed. Let us note that the following expressions are never defined 0 to the 0, or 0 to a negative integer power. These facts all hold for integers, which now can be negative as well as positive. And I have some good news.
Visually, these facts are exactly the same as they were for positive integer powers. Therefore, we already know these facts. Anyway, visually, they’re the same. Let us now turn to the graphs of powers with a negative integer. We already have one example. Remember the function 1 over x? We saw that it’s an odd function. We saw that it was decreasing on r plus, for example. Well, these sorts of characteristics carry over for other negative power functions as well. If n is odd, the graph will look rather similar. It’ll define an odd function. And if n is even, it’ll define an even function, which will be decreasing in the first quadrant. We’ll continue our discussion a moment with roots and radicals.