One of the things we've spent a lot of time talking about in previous courses and videos is the concept of functions and how various functions are going to give you different looking graphs. But one of the things that becomes really tricky is trying to remember all the various properties of functions, and that's what we're going to be talking about in this video. You're going to want to know these well because you're going to see all of these properties at some point in this course. So without further ado, let's get right into things. Now the first thing we're going to take a look at is what is called the maximum value of a function.
Now the maximum value is simply asking for where the highest point of your function is. And looking at this curve that I have, I can clearly see that the highest point occurs right there. So what this tells me is that for this value of x, we're going to get the highest output or y value that corresponds to that point. So what I can say is that this is going to be where our function is greater than all of the rest of these values for all x values. So this is telling me that this x value will give me the highest y value or the highest output for our function.
Now another thing we need to be familiar with is the minimum value. And the minimum is, in essence, asking you for the opposite of the maximum. So the maximum is the highest point, whereas the minimum is the lowest point. We can see the lowest point occurs right here. Now what this means is that fc, some point x we have here, is going to be less than or equal to the function for all x values.
So this x value is going to give us the lowest possible output when we put it into our function. So in essence, for both of these, you could think that this would be c right here for this function, and then this would be c right here for that function because we're looking at either the lowest value for minimum or the highest value for maximum. Now something else we need to be familiar with is the concept of functions that are increasing or decreasing. And there's a simple way that I like to think about this. There's an analogy I like to use.
What you can imagine is that there is a person, a little person, walking on our function, and they're going from left to right. Now notice if this person were walking from left to right, they would be traveling up the side of this function. So because this person is continuously going up, that's the trend of this function, we would say that this whole graph is going up, which means that we're dealing with an increasing function. Now, likewise, you can imagine we have the same person on a different graph over here, but this time, notice that the general trend is that this person is going down. Because the general trend is that this graph and this function is going down, that means we are now dealing with a decreasing function.
But what would happen if we had a situation where we had a person on our graph here, and they walked from left to right, and nothing seemed to really happen? They didn't really go up, and they didn't really go down. They kinda just kept going straight. Well, this is a case where we can say our graph is flat. Now whenever we have a flat graph, that means that we're dealing with a constant function.
Now the reason we call this a constant function is because you can imagine we just have one constant output. So notice here for this y value, let's pretend that this y value here is 2. If this y value was output 2, notice it's constantly going to be 2 at every point. We get a constant output, so for every possible input for x, we're going to get an output of 2 every single time that's not going to change. So this is why we call this a constant function.
Now another concept that's important to understand is the possible symmetry of a function. One of the symmetries that we'll see is something referred to as symmetry on the y-axis, and this would be an example of a function that's symmetric on the y-axis. And the reason why is because you can imagine taking this function and folding it like a piece of paper, where you crease it on the y-axis and fold it. Doing that would give you a symmetric graph that you would see for both sides of this function, so that's why we say it's symmetric on the y-axis. And whenever this happens, we say that we're dealing with an even function and that our function, if we input a negative x value, is going to give us the function output, the same thing as putting a positive x value in for our function.
And if you're not sure what this means, well, let me explain. So let's say we're looking at this point right here. And let's say that this x value is negative 3, and then the output at this point is 2. Well, if we go ahead and take positive 3 and put it into our function, so the positive x value, we're also going to get an output of 2. So notice both the negative x value and the positive x value both give you the same output as a y-value.
So that's a situation where you have an even function and you're dealing with a function that's symmetric on the y-axis. Now another symmetry that we need to be familiar with is symmetry about the origin. And this happens when you have a situation or a graph that looks like this. So the reason we say this is symmetric about the origin is it's symmetric only on this specific point. So you can imagine taking this whole graph and folding it first on the y-axis, and then taking this graph and folding it again on the x-axis.
This is going to give you a situation where the graph looks symmetric once you folded it twice. And you could also fold it on the x-axis first then the y-axis, and it would still look symmetric. So this is what we call an odd function. And what we say for these functions is that f -x is equal to - fx. Now understand what this means, well, let's imagine that we're dealing with this point right here.
Let's say that the x value that we're dealing with is negative 3, and the output for our y value is, say, negative 4. If that's our point right there and we go to this point up here, well, this point would be at positive 3, so that's the positive x value we're putting in. But then we would get positive 4 as our result. So notice here how we, in essence, got the polar opposite points when we went to the two sides of this graph. When we looked at the negative x value, we got a negative output.
When we went to the positive x value, we got a positive output. But notice that they're the same magnitudes, the same numbers. They're just one's all negative and one's all positive. That's the idea of dealing with an odd function. Now the last piece we're going to take a look at here is symmetry on the x-axis.
And symmetry on the x-axis would look something like this, where we have this kind of graph. Notice you if you were to take this graph and fold it or crease it like a piece of paper on the x-axis, and you were to fold it over, you would get this kind of symmetry on both sides when you do this fold. But there's something that might be kinda odd looking about this graph here. Notice how we have a situation where if we were looking at, say, this x value right here, this x value, let's just call it 2, we would get 2 outputs as a result. We never see that when it comes to functions, and that's because this is a situation where we are not dealing with a function.
So whenever you have this type of case where one x input gives you 2 y outputs, it's not a function. Another way you can picture this is you can say it doesn't pass the vertical line test. Because when testing if something's a function, if you can draw a vertical line that passes through more than one point on your graph, then it is not a function. So this is why we would say this is a situation where we do not have a function when dealing with something that is symmetric on the x-axis. So these are some of the common characteristics that you're going to see when dealing with graphs, most of these being functions, but this would be an example or an exception where we're not dealing with a function when we're symmetric on the x-axis.
So hope you found this video helpful. Let's go ahead and get some more practice.
