Welcome back everyone. We're gonna continue with our module on functions. And here, we're really going to focus our attention to functions from the real line to the real line. And the idea of today's lecture is the idea of an increasing or decreasing function. These are two special classes of functions. It's important to understand that most functions don't fall onto either, but we need to understand which one's fall into which class. So if we look around the left, we have three functions. We have the graph of f_of_x, which is the red graph. We have a graph of g_of_x, which is the blue graph. And we have the graph of h_of_x, which is the yellow graph. You'll notice f_of_x seems to go up as you move along the x axis and you look at where you are on the red graph, you're always climbing. Here's the graph of g has the opposite and always falling as I move along. Whereas the graph of h has a falling period and then an increasing period. We say that f is a strictly increasing function. We say that g is a strictly decreasing function. And turns out that h is neither. As we often do as mathematicians, were not satisfied just with visual intuitive definitions, we like to be annoying and write down some symbols. So over here on the right I've done that. Let's look at what these symbols mean, and let's look at how to translate those intuitive definitions we already see. Remember, the idea is that the red should be strictly increasing, the blue should be strictly decreasing, h is neither. To the definitions. So let's let f be a function from real line to real line and f can stand in for any of these. We say that f is strictly increasing if- and this condition is tricky to parse. Whenever two inputs, a and b, have a relationship that a is less than b, it must be that the output f_of_a, f_of_b have the same relationship. Let's check that f is strictly increasing. For example, if this is the point a and this is the point b, notice that a and b are less, a is less than b. So look up here, there is f_of_a on the graph, there's f_of_b. And notice, they have the same order relationship. And I didn't just get lucky and pick those two. Those could've been anywhere along the x axis. It turns out that f, put that in red, f is strictly increasing. We say that f is strictly decreasing if whenever a is less than b, the order of relationship flips the output,so f_of_a is greater than f_of_b. Turns out the blue guy, that's true, but if you look at the blue guy, here is a less than b, so if they were to hit the graph, there's g_of_a, there's g_of_b. You know they're getting closer to each other, g_of_b is actually less than g_of_a. And that's true no matter where I picked a and b. You can check for yourself that. So let's write that in here. G is strictly decreasing. On the other hand, I'll just give you the answer right away, h is neither. Can you convince yourself why that's true? If I took a and b say over here, and I looked at where a hits the graph and I looked at where b hits the graph, notice that the order of relationship flips, a is less than b, but h_of_a is greater than a_of_b. On the other hand, if I took a and b where we have them over here and I looked at where they hit the graph, h_of_a is actually less than h_of_b. So there's no consistency here. Okay, so we've seen some picture examples. Let's write down some actual functions with formulas and lets figure out whether they're strictly increasing or decreasing. And a little secret here as I'm also going to show you some functions we haven't seen before. So when we consider the function f(x) = 2 to the x. Then we considered g(x) = 3 to the minus x, and one you've seen before which is h(x) = x squared. Now let's figure out which of these are strictly increasing, which are strictly decreasing, which are neither. Over here, our axis. And let's first start by drawing the graph of f(x) = 2x to the X. You may not seen this before, this is called an exponential function. And the ideas that up here, x is in the x spot. Let's figure out a table of values and see what that might look like. If x is zero, this is x and this is f(x), if x is zero, then we have two to the zero, which is one. and here's the 0.01. X is one, we have two to the one equals two, x is two, we have two to the two which is four, x is three, two cube = eight. Shoots all way up there. X is minus 1, 2 to the minus 1 is 1 over 2 to the 1 is 1/2, and so on. So the graph actually end up looking like this sort of, and on, and on. So there's f(x) = 2 to the x. That's the graph f(x) = 2 the X. Turns out that's strictly increasing, as you could probably tell. Okay, in blue, let's figure out the graph of g(x) = 3 to the minus x. So let's make a own table of values. here's x, here's g(x), there's 0, so g(0) would be 3 to the 0 is 1. So it's here. 1 will be 3 to the minus 1. 1/3 and we get the pattern, it's going to go quite steeper, and up like that and down like that. So, g(x) is strictly decreasing. Okay. Let's draw the graph of h(x) = x squared in yellow. So let's see that should go through the point 1, 1 and 1 minus 1, looks about like that. Now h is neither strictly increasing nor strictly decreasing. Sometimes is going up and sometimes is coming down. Here's a statement which you should intuitively understand what this means, but h is strictly increasing on the interval from zero to infinity. In other words, if I restrict myself only the things in this interval here, and I only use those in inputs, h satisfies the definition of strictly increasing. I plug in two points in that- it goes up, and h is strictly decreasing on the interval from minus infinity to zero and you convince your self. So that's interesting. OK Let's understand what this might have to do with say, real world examples. So we've seen some examples of functions that I've just made up, here are some examples that might make more sense in the real world, a world we might have increasing functions and decreasing functions. On the left, let's imagine plotting over here years since birth of a typical child, and over here, lets plot height. We won't really commit to scale of units, will just get a sense of the general shape. So then of course nothing to the left of the y axis matters, because they aren't negative years since birth. This will almost entirely be an increasing function. Right, you'll start at year one, we'll measure you about here and you'll go up, you go up really violently and then just start to level off. Right around here that's probably about 17, you'll stay sort of stable for a long, long time, maybe a little bit of growth, and you might get down a little bit sadly if you stoop. So that's a function which is increasing for a long bit, then it's sort of flat and then goes down. As always the mathematical notions will be much more precise and what happens in real life, wouldn't be a smooth curve it would depend on the measurements of the doctor's office. Over here, suppose we have years since purchasing a car, and over here we have value of a car. Without even drawing it, you probably predict that's going to be a decreasing function. Right, here's year one, it's one of my cars, it's going to start about a thousand dollars. Perhaps that's revaluing me, and then it's going to decrease. Soon as you drive it off the lot goes down a little bit and it's going to plateau somewhere near the bottom. OK, we're going to close with just a simple visual test to tell whether or not our function is increasing or decreasing. Notice the red graph is a graph of the strictly increasing function. The blue graph is a graph of a function which is neither strictly increasing nor strictly decreasing. Really simple way to do it is called the horizontal line test. You may notice, if I draw any on a line here in green, it hits the right graph exactly once, no matter where I draw the line. Which makes sense, right. Because that's the value that's hit that one time. It could never, ever return to that value, because if it did- the graph would have had to bend back down. So in other words there's the value I hit, and I don't get to hit that value again because the graph is gone. See you later, it's all on the train. On the other hand you'll notice blue, there are many horizontal lines which hit the graph twice. This horizontal line here hit here. Now blue goes down, again we've left the line forever but it starts to come back up again and hits the graph. And so that's really the horizontal line test. The function is strictly increasing or strictly decreasing if whenever you graph it every single horizontal line has to intersect exactly once.
