Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Functions. A function is even if X and negative X are in the domain and F and negative X = F of X. This F of X is really our Y value. So we're thinking about our Y value for when X is put in has to be the same Y value as when negative X is put in. So if I thought about when X IS1Y is 2, then when X is -1 the Y value has to be the same or two. Let's say when X is 2 we get AY value of four when -2 for X we need the same Y value out as when we had the +2 or 4. Let's make this into a parabola and say when F is 0 is 0. So if we look at this graph, we realize we have to have it being symmetric about the Y axis. So even functions are symmetric about the Y axis. Now let's look at an odd function. X and negative X have to be in the domain again, and this time the Y value at negative X is going to be the opposite of the Y value at the X. You could also think of this as the opposite of F of negative X equaling F of X. Those are really saying that the same thing. If you think about multiplying each side by a -1, this is an equivalent. So let's look at if X is 1 and Y is 2. Now when we use -1, we want the opposite of two, which would be -2. If we use X as two and Y as 4, then at -2 we want the opposite of it, which would be -4. Once again, let making it into a function, let's use F as zero being 0. So here we get a graph that looks something like that, and for a function to be odd, it's got to be symmetric about the origin. A function is increasing on an open interval. If for all X in the interval X1 is less than X2, then the Y value at X1 is less than the Y value at X2. If we look at this on a graph, the X1 has to be less than F2 or to the left of X2. The F of X1 has to be less than the F of X2 and this has to be true for all the values of the function for it to be increasing. Now decreasing very similar a function is decreasing on an open interval. If for all X in the interval X1 less than X2, then the Y value at X1 has to be greater than the Y value at X2. If we think about this one pictorially, X1 had to be to the left of X2 but now the F of X1 had to be bigger than the F of X2. So here and here. And this has to be true for all the XS in that open interval. The last topic is a function is constant if the F of X values are the same. So now no matter what my X values are, the F of X values are going to be the same throughout that interval. A local maximum occurs in an open interval if for every X in the interval, F of X is less than or equal to F of C where C is in the interval F of C is called a local maximum value of F. An absolute maximum occurs when F is defined on some interval and F of X is less than or equal to some F of U for all X in the interval. So if we look at a graph and we look at various intervals, if we look at the open interval from here to here, here's my Y value that goes with this point. Here is the Y value that goes with this point. So we have some absolute maximum here, but it's also a local because we're looking at that single interval, that open interval here. If we changed our interval to be a closed interval, let's say we go from here all the way down to here. Now we have a local Max here because there's any old open interval that I could use, and that'd be the highest point of all of them on this interval. This one, however, would be an absolute because it's higher than all of the others on the entire interval. We have the same concept with minimums. So a local minimum occurs in an open interval if for every X in the interval, F of X is greater than or equal to F of C where F of C is in the interval or where C is in the interval. F of C is called a local min. An absolute min occurs when F is defined on some interval and F of X greater than or equal to F of V for all X in the interval. The difference here basically is that this one has to be an open interval, so it can be a very small interval, and this one has to be for the entire interval that it's defined on. So if we look here, just throwing in something really quick, if we look at this interval here, this is going to be a local min and also an absolute min if we're looking at just on this particular interval, because somewhere I could put an open interval for the local min. And then on the entire interval here, this is still the lowest Y value. If we wanted to make our interval a little bigger, let's say we're going from here all the way out to here, then this one would be a local min, but this one down here, because it's lower Y value, would be the absolute min. We also have extreme value theorem, and what the extreme value theorem says is if there's a function that's continuous whose domain is a closed interval, IE the two endpoints are solid, they're defined, then F has an absolute Max and an absolute minimum somewhere on AB. So if we look at some graph closed, there's got to be an absolute Max and an absolute min somewhere. There would be my absolute Max, here would be an absolute min. Now it doesn't tell me anything about locals, but in this case right here would be a local min because it's smaller than everything else that's around it. Thank you and have a wonderful day.