College Algebra: Find Function Increase Intervals

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When you look at the graph of a function, it's going up and down, it's quite exotic looking. Well, this raises the question, where is the function sort of going down, sort of falling, and when is the function sort of increasing? I mean, you can imagine a function that looks sort of like this. A function sort of goes up and goes down and goes up and goes down, and so forth. I want to tell you how you identify where the function is falling and where the function is rising, or whether the function is decreasing or increasing.
A function is decreasing when the function drops as the x values get larger. Now, this is sort of a teeny point, which really has to be clear, though. You see, when you think about the real x line here, the axis, you know that the numbers increase when you go off to the right. So as you go off to the right, the question is what is the function doing? Now, look what happens. We start way up here; the function is falling as you go to the right. So this is a region where the function is decreasing. But then the function levels off here and starts to go up. So now this is an area where the function is increasing. Then the function starts to decrease again, and then here the function starts to increase. So this is a function where the function first decreases then increases, then decreases, then increases.
Okay, fine. What's the big deal? Well, there's not really a big deal; however, let's take a look at some examples just to sort of drive this point home. Here's a parabola-looking-like curve. You can see that the function is first decreasing and then increasing. So where's the region where the function is decreasing? It would be all the way out from minus infinity all the way up to zero. Those are the x values for which the function is decreasing. But then once you go from zero out, the function starts to increase again. Now why do I make a big stink about this? I make a big stink because a lot of the time when we draw pictures we put these arrows like this to indicate that the thing keeps going. But if you look at the arrow you might say, "Hey, that's like a one-way street. I should be going this way." But if you're looking at it that way you might be saying, "Hey, wait a minute. It looks like I'm going up. I'm increasing." Avoid that temptation to think that and always remember that to see if a function is increasing or decreasing you always start on the left and then go to the right and ask what would happen if I put a little ball there. If I put a little ball there it would roll down, and then here you'd have to push it up. Even though the arrow is here, this is still decreasing and then increasing.
Okay, let's try some more. This is a straight line and what do you see here? Well, here you see it's just going down, so this is decreasing, and in fact, you can see the rate of decrease is always constant. In fact, this is foreshadowing for what's going to be called slope, the slope of a line, the bending-ness of a line. It's very straight, but it's going down. So this is decreasing everywhere. Here's a function that's sort of like a tee-pee-like function. This actually is a function that we saw, at least something that's similar to it, a long time ago when we looked at some basic functions, it's like the absolute value function. The absolute value function, of course, has the sharp v-like look. This is not quite that. This is like a tee-pee look, which means it's probably like the negative of an absolute value. Somehow it's flipped. And we'll talk about that later.
Anyway, the point is, what's going here is you can see that the function is first increasing until it gets to x = 2, and then after x = 2 in this range, the function starts to fall. So this function is increasing for x between negative infinity all the way up to 2, because in this range, the function is climbing, and then from 2 out to infinity the function is falling. So, first increasing, and then decreasing.
This is a function that just is this line that's parallel to the x-axis, but two units down, so this is actually -2 here. And what's going on here? Well, here what's happening is we have the function being constant. It's neither increasing nor decreasing. It's constant. It's unchanging. Its height is always at -2, so in fact, we call this constant. This is a constant function.
Here's a kinky curve, because it sort of does something and then it kinks and does this. So what's going on here? Well, it's constant everywhere up to x = 0. So from negative infinity to zero it's constant, but then, from here on, increasing. So increasing here, from zero to positive infinity, increasing. From negative infinity to x = 0, constant. See, piece of cake once you get these things down.
Here's a function that's very kinky. It's this, then kinks, then kinks again. So then what do we see here? From negative infinity to x = 0 we see it's constant, then we see from 0 to 3 it's decreasing, and then from 3 onward it's constant again. So this is constant to x = 0, then from 0 to x = 3 it's decreasing, and then from 3 to infinity it's constant. Even more kinky. You can see we're getting kinkier as we go along here, not surprising. We have a kink here, a kink here, a kink here, and a kink here. What do we have? Decreasing, increasing, decreasing, increasing. And so how does that go exactly? The function is decreasing from negative infinity up to -3, then from -3 all the way to x + 0 we have an increasing thing, and then from x = 0 to x = 3 it's decreasing again, and then from x = 3 to infinity it increases one last time. So it decreases, increases, decreases, increases. Great.
What about this one? It's another line. But now the line is actually going up as you go to the right, so this is increasing everywhere. Increasing everywhere. Here is some more kinky stuff for those people who get into this. A little kinky here. It's decreasing, then it's constant for a little teeny bit, but it is constant here, and then increasing again. So what do we have? We have decreasing from -x = minus infinity all the way up to zero. It's decreasing all on this region. Then from x = 0 to x = 1 it seems to be constant, and then from x = 1 onward the thing is increasing. So decreasing to x = 0, from x = 0 to x = 1 it's constant, and then from 1 to infinity, increasing. Great.
One last one. This is sort of like a cubic thing. This has like an x-cubed in it probably. And what do I see? Well, even though it sort of slows down a little bit and then goes up, we actually see it's always increasing. So this actually is increasing--climb, climb, climb, climb, climb, climb, climb--slow climb, but still climbing, slow climb, still climbing, slow climb, still climbing, and now really fast climbing. This is increasing everywhere. So the notion of increasing, decreasing, and then constant. And, of course, also, where you change from increasing to deceasing we have either a max or a min. So, in fact, if you have something that has this basic flavor--and I'll try to put everyone on, even a sharp point--then what you see is whenever you change from either increasing to decreasing, wherever that change happens, we're going to have a point where the function sort of changes. In this case we have a max. So if you go from increasing to decreasing you must have a max, right? You're climbing and then falling; you must have a high point.
Similarly, if you go from a decreasing to an increasing, there you must have a low point. So these are called maxes, and this is called a min. So if you go from an increasing and then to a decreasing you must have another max. If you go from a decreasing to an increasing you must have another min, and so forth. So you can find all the maxes and mins by looking at where the function changes from being increasing to decreasing or decreasing to increasing, and you have either a max or min. Great. You can identify these special points on your favorite function. Enjoy.
Relations and Functions
Working With Functions
Determining Intervals Over Which a Function is Increasing Page [2 of 2]